# Formulas for Generalized Two-Qubit Separability Probabilities

Authors: Paul B. Slater
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

To begin, we find certain formulasQ(k,α)=G1k(α)G2k(α), for k=-1,0,1,…,9. These yield that part of thetotal separability probability, P(k,α), for generalized (real, complex, quaternionic, etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality ρPT>ρ holds. Here ρ denotes a 4×4 density matrix, obtained by tracing over the pure states in 4×(4+k)-dimensions, and ρPT denotes its partial transpose. Further, α is a Dyson-index-like parameter with α=1 for the standard (15-dimensional) convex set of (complex) two-qubit states. For k=0, we obtain the previously reported Hilbert-Schmidt formulas, with Q(0,1/2)=29/128 (the real case), Q(0,1)=4/33 (the standard complex case), and Q(0,2)=13/323 (the quaternionic case), the three simply equalling P(0,α)/2. The factors G2k(α) are sums of polynomial-weighted generalized hypergeometric functions pFp-1, p≥7, all with argument z=27/64=(3/4)3. We find number-theoretic-based formulas for the upper (uik) and lower (bik) parameter sets of these functions and, then, equivalently express G2k(α) in terms of first-order difference equations. Applications of Zeilberger’s algorithm yield “concise” forms of Q(-1,α), Q(1,α), and Q(3,α), parallel to the one obtained previously (Slater 2013) for P(0,α)=2Q(0,α). For nonnegative half-integer and integer values of α, Q(k,α) (as well as P(k,α)) has descending roots starting at k=-α-1. Then, we (Dunkl and I) construct a remarkably compact (hypergeometric) form for Q(k,α) itself. The possibility of an analogous “master” formula for P(k,α) is, then, investigated, and a number of interesting results are found.


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## Body

## 1. Introduction

In a previous paper [1], a family of formulas was obtained for the (total) separability probabilities P(k,α) of generalized two-qubit states (N=4) endowed with Hilbert-Schmidt (k=0) [2] or more generally random induced measure [3, 4]. In this regard, we note that the natural, rotationally invariant measure on the set of all pure states of a N×K composite system (k=K-N) induces a unique measure in the space of N×N mixed states [3, eq. (3.6)]. Further, α serves as a Dyson-index-like parameter [5, 6], assuming the values 1/2,1,2 for the two-rebit (N=4), two-qubit (standard/complex), and two-quaterbit states, respectively.The concept itself of a “separability probability,” apparently first (implicitly) introduced by Życzkowski et al. in their much cited 1998 paper [7], entails computing the ratio of the volume—in terms of a given measure [8]—of the separable quantum states to all quantum states. Here, we first examine a certain component Q(k,α) of P(k,α). This informs us of that portion—equalling simply P(k,α)/2 in the Hilbert-Schmidt (k=0) case [9]—for which the determinantal inequality ρPT>ρ holds, with ρ denoting a 4×4 density matrix and ρPT denoting its partial transpose. By consequence [10] of the Peres-Horodecki conditions [11, 12], a necessary and sufficient condition for separability in this 4×4 setting is that ρPT>0. The nonnegativity condition ρ≥0 itself certainly holds, independently of any separability considerations. So, the total separability probability can clearly be expressed as the sum of that part for which ρPT>ρ and that for which ρ>ρPT≥0. The former quantity will be the one of initial concern here, the ones the formulas Qk,α will directly yield.The complementary quantity that for whichρ>ρPT≥0 can, in the most basic cases of interest, be readily obtained from thetotal separability probability formulas Pk,α reported in [1], which took the form(1)Pk,α=1-Fk,α,where for integral and half-integral α,(2)Fk,α=pαkGk,α,with(3)Gk,α≔4kΓk+3α+3/2Γ2k+5α+2Γ1/2Γ3k+10α+2.Here, for integral α, pαk is a polynomial of degree 4α-2 with leading coefficient 28α+1/2α-1!.In [1], certain α-specific formulas (α=1,2,…,13 and 1/2, 3/2, 5/2, and 7/2) had been derived (and we have since continued the integral series to α=73). Most notably [1, eq. (3)],(4)Pk,1=1-34k+32kk+7+25Γk+7/2Γ2k+9πΓ3k+13.Here P(k,1) denotes the total separability probability of the (15-dimensional) standard, complex two-qubit systems endowed with the random induced measure for k=K-4. Further, in the two-quater[nionic]bit setting [1, eq. (4)],(5)Pk,2=1-4k+6kk2kk+21+355+1452+2430Γk+13/2Γ2k+153πΓ3k+22.Also, for the two-re[al]bit scenario [1, eq. (5)],(6)Pk,12=1-4k+18k+15Γk+2Γ2k+9/2πΓ3k+7.Tables 1, 2, and 3 in [1] reported for k=0,1,…,8, the, in general, rather simple fractional separability probabilities P(k,α) yielded by these three formulas.By way of example, we first note that formula (4) yields P(1,1)=61/143. Then, since we will find from our analyses below that Q(1,1)=45/286, we can readily deduce that the corresponding (complementary) separability probability corresponding to the inequalities ρ>ρPT≥0, for this k=1, α=1 scenario, is equal to P(1,1)-Q(1,1)=7/26=61/143-45/286.Let us further observe that, for the Hilbert-Schmidt (k=0) case, strong evidence has been presented [9] that, for the two-rebit, two-qubit, and two-quaterbit cases, the apparent total separability probabilities P(0,α) of 29/64,8/33, and 26/323, respectively, are equally divided between the two forms of determinantal inequalities (cf. [13]). Lovas and Andai have recently formally proven this two-rebit result and presented an integral formula; they hope to similarly yield the two-qubit proportion [14]. (These “half-probabilities,” remarkably, are also the corresponding separability probabilities of theminimally degenerate states [13], those for which ρ has a zero eigenvalue.) For k>0, however, our analyses will indicate that equal splitting is not, in fact, the case. Greater separability probability is associated with the ρ>ρPT≥0 inequality than ρPT>ρ. Thus, in the k=1, α=1 instance just discussed, we do have 7/26>45/286. (On the other hand, if k=-1, then necessarily ρ=0, so all the total separability probability P(-1,α) must, it is clear, be assigned to the ρPT>ρ component. That is, Q-1,α=P-1,α. Observations of this nature should help in the further understanding of the intricate geometry of the generalized two-qubit states endowed with random induced measure (cf. [15]).


## 2. Procedures

### 2.1. Previous Analyses

To obtain the new formulasQ(k,α) to be presented here for the separability probability amounts for which ρPT>ρ holds, we first employed—as in our prior studies [1, 9, 16, 17]—the Legendre-polynomial-based probability density-approximation (Mathematica-implemented) algorithm of Provost [18] (cf. [19]). In this regard, we utilized the previously obtained determinantal moment formula [1, eq. (6)] [9, sec. II] (cf. [20]) (7)ρkρPT-ρnρk=-1nαnα+1/2nn+2k+2+5αn24nk+3α+3/2n2k+6α+5/22n×F43-n2,1-n2,k+1+α,k+1+2α1-n-α,12-n-α,n+2k+2+5α;1,where the variable k has the same sense as indicated above, in equalling K-4, and the bracket notation indicates averaging with respect to the random induced measure. Here, ρk=1k3/2k2k5/2k/104k, where the Pochhammer (rising factorial) notation is employed.On the other hand in [1], a second companion moment formula [16, sec. X.D.6], (8)ρPTn=n!α+1n2α+1n26n3α+3/2n6α+5/22n+-2n-1-5αnαnα+1/2n24n3α+3/2n6α+5/22nF54-n-22,-n-12,-n,α+1,2α+11-n,n+2+5α,1-n-α,12-n-α;1,had been utilized for density-approximation purposes with the routine of Provost, with the objective of finding thetotal separability probabilities P(k,α), associated with the Peres-Horodecki-based inequality ρPT>0. (These moment formulas had been developed in [16], based on calculations solely for the two-rebit [α=1/2] and two-qubit [α=1] cases. However, they do appear, as well, remarkably, to apply to the two-quater[nionic]bit [α=2] case, as reported by Fei and Joynt in a highly computationally intensive Monte Carlo study [21]. No explicit formal extension of the Peres-Horodecki positive-partial-transposition separability conditions [11, 12] to two-quaterbit systems seems to have been developed, however [cf. [22–24]]. The value α=4 corresponds, presumably it would seem, to anoctonionic setting [25, 26].)


### 2.2. Present Analyses

Here, contrastingly (“dually”) with respect to the approach indicated in [1], we will find k-specific formulas (k=-1,0,1,…,9) as a function of α, that is, Q(k,α), for the indicated one (ρPT>ρ) of the two component determinantal inequality parts of ρPT>0. We utilized an exceptionally large number (15,801) of the first set of moments above in the routine of Provost [18], helping to reveal—to extraordinarily high accuracy—the rational values that the corresponding desired (partial) separability probabilities Q(k,α) strongly appear to assume. Sequences (α=1,2,…,30,…) of such rational values, then, served as input to the FindSequenceFunction command of Mathematica, which then yielded the initial set of k-specific (hypergeometric-based) formulas for Q(k,α). (This apparently quite powerful [but “black-box”] command of which we have previously and will now make copious use has been described as attempting “to find a simple function that yields the sequence when given successive integer arguments.” It can, it seems, succeed too, at times, for rational-valued inputs, and perhaps even ones of a symbolic nature.) We, then, decompose Q(k,α) into the product form G1k(α)G2k(α).


## 2.1. Previous Analyses

To obtain the new formulasQ(k,α) to be presented here for the separability probability amounts for which ρPT>ρ holds, we first employed—as in our prior studies [1, 9, 16, 17]—the Legendre-polynomial-based probability density-approximation (Mathematica-implemented) algorithm of Provost [18] (cf. [19]). In this regard, we utilized the previously obtained determinantal moment formula [1, eq. (6)] [9, sec. II] (cf. [20]) (7)ρkρPT-ρnρk=-1nαnα+1/2nn+2k+2+5αn24nk+3α+3/2n2k+6α+5/22n×F43-n2,1-n2,k+1+α,k+1+2α1-n-α,12-n-α,n+2k+2+5α;1,where the variable k has the same sense as indicated above, in equalling K-4, and the bracket notation indicates averaging with respect to the random induced measure. Here, ρk=1k3/2k2k5/2k/104k, where the Pochhammer (rising factorial) notation is employed.On the other hand in [1], a second companion moment formula [16, sec. X.D.6], (8)ρPTn=n!α+1n2α+1n26n3α+3/2n6α+5/22n+-2n-1-5αnαnα+1/2n24n3α+3/2n6α+5/22nF54-n-22,-n-12,-n,α+1,2α+11-n,n+2+5α,1-n-α,12-n-α;1,had been utilized for density-approximation purposes with the routine of Provost, with the objective of finding thetotal separability probabilities P(k,α), associated with the Peres-Horodecki-based inequality ρPT>0. (These moment formulas had been developed in [16], based on calculations solely for the two-rebit [α=1/2] and two-qubit [α=1] cases. However, they do appear, as well, remarkably, to apply to the two-quater[nionic]bit [α=2] case, as reported by Fei and Joynt in a highly computationally intensive Monte Carlo study [21]. No explicit formal extension of the Peres-Horodecki positive-partial-transposition separability conditions [11, 12] to two-quaterbit systems seems to have been developed, however [cf. [22–24]]. The value α=4 corresponds, presumably it would seem, to anoctonionic setting [25, 26].)


## 2.2. Present Analyses

Here, contrastingly (“dually”) with respect to the approach indicated in [1], we will find k-specific formulas (k=-1,0,1,…,9) as a function of α, that is, Q(k,α), for the indicated one (ρPT>ρ) of the two component determinantal inequality parts of ρPT>0. We utilized an exceptionally large number (15,801) of the first set of moments above in the routine of Provost [18], helping to reveal—to extraordinarily high accuracy—the rational values that the corresponding desired (partial) separability probabilities Q(k,α) strongly appear to assume. Sequences (α=1,2,…,30,…) of such rational values, then, served as input to the FindSequenceFunction command of Mathematica, which then yielded the initial set of k-specific (hypergeometric-based) formulas for Q(k,α). (This apparently quite powerful [but “black-box”] command of which we have previously and will now make copious use has been described as attempting “to find a simple function that yields the sequence when given successive integer arguments.” It can, it seems, succeed too, at times, for rational-valued inputs, and perhaps even ones of a symbolic nature.) We, then, decompose Q(k,α) into the product form G1k(α)G2k(α).


## 3. Common Features of thek-Specific Formulas Qk,α

For eachk=-1,0,1,…,9, the FindSequenceFunction command yielded what we can consider as a large, rather cumbersome (several-page) formula, which we denote by Q(k,α). These expressions, in fact, faithfully reproduce the rational-valued (separability probability) sequences that served as input. This fidelity is indicated by numerical calculations to apparently arbitrarily high accuracy (hundreds of digits). (The difference equation results below [Section 5] will provide a basis for our observation as to the rational-valuedness [fractional nature] of these separability probabilities.)In Figure1, we show plots of the formulas Q(k,α) obtained over the range α∈[1,10], for k=-1,…,9. For fixed α, we have Q(k1,α)>Q(k2,α), if k1>k2. In Figure 2, we show a companion plot, exhibiting strongly log-linear-like behavior, for log⁡Q(k,α).Figure 1
Plots of the separability probability formulasQ(k,α) over the range α∈[1,10], for k=-1,…,9. For fixed α, we have Q(k1,α)>Q(k2,α), if k1>k2.Figure 2
Plots oflog⁡Q(k,α) over the range α∈[1,10], for k=-1,…,9. For fixed α, log⁡Q(k1,α)>log⁡Q(k2,α), if k1>k2.


### 3.1. DistinguishedF76 Function with 2 as an Upper Parameter in Q(k,α)

In each of the elevenk-specific formulas Q(k,α) obtained, there is a distinguished F76 generalized hypergeometric function, with the (“omnipresent,” we will find) argument of z=27/64=(3/4)3 (cf. [27] [28, Ex. 8.6, p. 159]), having 2 as one of the seven upper parameters (cf. [17]).


#### 3.1.1. The Six Lower Parameters

The lower (bottom) six parametersbik, i=1,…,6, of the 7F6 function conform for all eleven cases to the simple linear rule(9)b1k,b2k,b3k,b4k,b5k,b6k=α+2k5+2310,α+2k5+52,α+2k5+2710,α+2k5+2910,α+2k5+3110,α+k+3.The six entries are summed to 6α+3k+33/2.


#### 3.1.2. The Six Upper Parameters

The six upper parameters (aside from the seventhk-invariant constant of 2 already indicated), {u1k,u2k,u3k,u4k,u5k,u6k}, can be broken into one set of two (the numerical parts summing to integers), incorporating consecutive fractions having 6’s in their denominators, and one set of four (the numerical parts also summed to integers), incorporating consecutive fractions having 5’s in their denominators. For the set of two, the smaller of the two upper entries abides by the rule(10)u1k=α+164k3+2k+13+11,where the (integer-valued) floor function is employed, while the larger entry is given by(11)u2k=α+162k3+4k+13+13.For integral values of k, the same values of ui1 and ui2 are yielded by the interpolating functions(12)α+1186k+2cos⁡2πk3+2cos⁡4πk3+29,α+1186k-3sin⁡2πk3+3sin⁡4πk3+cos⁡2πk3+cos⁡4πk3+37,respectively.Fork=1, for illustrative purposes, application of the two rules yields α+11/6,α+13/6, and for k=5, we have α+19/6,α+23/6. (We have noted that u1k+u2k-2α=k/3+k+1/3+4 is an integer. The sequence of these integers—for arbitrary integer or half-integer values of α—is found in the On-Line Encyclopedia of Integer Sequences [https://oeis.org/ol.html] as A004523 [“two even followed by one odd”] and as A232007 [“maximal number of moves needed to reach every square by a knight from a fixed position on an n×n chessboard, or −1 if it is not possible to reach every square”].)For the complementary set of four upper parameters of the7F6 function, the entries in order of increasing magnitude are expressible as(13)u3k=α+153k-45+2k-35+2k-25+3k-15+16,u4k=α+153k-45+2k-35+k-25+4k-15+17,u5k=α+152k-45+3k-35+k-25+4k-15+18,u6k=α+152k-45+3k-35+k-25+4k-15+19.For k=1, for illustrative purposes, application of these four rules yields α+9/5,α+11/5,α+12/5,α+13/5, and for k=5, we have α+16/5,α+17/5,α+18/5,α+19/5. For arbitrary k, the sum of the four terms under discussion minus 4α is an integer, namely, 2k-4/5+2k-3/5+(k-2)/5+3k-1/5+14. Further, let us note that, for integral values of k, u3k has values (14)1250502k+7+50-105-3sin⁡25π1-2k+2sin⁡4πk5-2sin⁡6πk5-3sin⁡15π-6πk+2sin⁡15π-4πk-3sin⁡253πk+π+sin⁡154πk+π+2sin⁡156πk+π+105+5-2sin⁡25π1-4k-2sin⁡2πk5+2sin⁡8πk5-2sin⁡25πk+1-3sin⁡15π-2πk-sin⁡254πk+π+sin⁡158πk+π+3cos⁡1104πk+π.


### 3.2. DistinguishedF76 Function with 1 as an Upper Parameter in Q(k,α)

Eachk-specific formula Q(k,α) we have found also incorporates a second F76 function (again with argument z=27/64, which is, to repeat, invariably the case throughout this paper), having all its thirteen parameters simply equalling 1 less than those in the function just described. (A basic transformation exists [consulting the HYP manual of C. Krattenthaler, available at http://mathematik.univie.ac.at], allowing one to convert the thirteen [twelve α-dependent parameters, plus 1] of this F76 function [i.e., add 1 to each of them] to those thirteen of the first distinguished F76 previously described, plus other terms.)


### 3.3. Themk Remaining Fpp-1 Functions, p=8,…,8+mk-2, in Q(k,α)

Now, in addition to the two distinguishedF76 functions just presented, there are mk more hypergeometric functions Fpp-1, p>7, for each k, where(15)m-1,m0,m1,m2,m3,m4,m5,m6,m7,m8,m9=3,5,5,6,6,7,9,8,10,10,10.Each of these additional functions possesses, to begin with, the same seven upper parameters (that is, 2, plus those six indicated in (10), (11), and (13)) and the same six lower parameters (9), as in the first 7F6 function detailed above (Section 3.1). Then, the seven upper parameters are supplemented by from 1 to mk 2’s, and the six lower parameters supplemented by 1 to mk 1’s.


### 3.4. Largeα-Free Terms Collapsing to 0

We now point out a rather remarkable property of the formulas forQ(k,α) yielded by the FindSequenceFunction command. If we isolate those (often quite bulky) terms that do not involve any of the mk+2 hypergeometric functions for each k already described above, we find (to hundreds of digits of accuracy) that they collapse to zero. These terms, typically, do contain hypergeometric functions similar in nature to those described above, but with the crucial difference that the Dyson-index-like parameter α doesnot occur among their upper and lower parameters. Thus, we are left, after this nullification of terms, with formulas Q(k,α) that are simply sums of mk+2 polynomial-weighted Fpp-1 functions (of α), with p=7,7,8,…,7+mk.


### 3.5. Summary

To reiterate, for eachk, our formulas for Q(k,α) all contain a single function of the form(16)F762,uk,u2k,u3k,u4k,u5k,u6k;bk,b2k,b3k,b4k,b5k,b6k;2764.There is another distinguished single F76 function, with all its thirteen parameters being one less. Also there are mk additional functions, i=1,…,mk,(17)F7+i6+i2,2,…,uk,u2k,u3k,u4k,u5k,u6k;1,…,bk,b2k,b3k,b4k,b5k,b6k;2764,with the number of upper 2’s running from 2 to mk+1 and the number of lower 1’s, simultaneously running from 1 to mk.


## 3.1. DistinguishedF76 Function with 2 as an Upper Parameter in Q(k,α)

In each of the elevenk-specific formulas Q(k,α) obtained, there is a distinguished F76 generalized hypergeometric function, with the (“omnipresent,” we will find) argument of z=27/64=(3/4)3 (cf. [27] [28, Ex. 8.6, p. 159]), having 2 as one of the seven upper parameters (cf. [17]).


### 3.1.1. The Six Lower Parameters

The lower (bottom) six parametersbik, i=1,…,6, of the 7F6 function conform for all eleven cases to the simple linear rule(9)b1k,b2k,b3k,b4k,b5k,b6k=α+2k5+2310,α+2k5+52,α+2k5+2710,α+2k5+2910,α+2k5+3110,α+k+3.The six entries are summed to 6α+3k+33/2.


### 3.1.2. The Six Upper Parameters

The six upper parameters (aside from the seventhk-invariant constant of 2 already indicated), {u1k,u2k,u3k,u4k,u5k,u6k}, can be broken into one set of two (the numerical parts summing to integers), incorporating consecutive fractions having 6’s in their denominators, and one set of four (the numerical parts also summed to integers), incorporating consecutive fractions having 5’s in their denominators. For the set of two, the smaller of the two upper entries abides by the rule(10)u1k=α+164k3+2k+13+11,where the (integer-valued) floor function is employed, while the larger entry is given by(11)u2k=α+162k3+4k+13+13.For integral values of k, the same values of ui1 and ui2 are yielded by the interpolating functions(12)α+1186k+2cos⁡2πk3+2cos⁡4πk3+29,α+1186k-3sin⁡2πk3+3sin⁡4πk3+cos⁡2πk3+cos⁡4πk3+37,respectively.Fork=1, for illustrative purposes, application of the two rules yields α+11/6,α+13/6, and for k=5, we have α+19/6,α+23/6. (We have noted that u1k+u2k-2α=k/3+k+1/3+4 is an integer. The sequence of these integers—for arbitrary integer or half-integer values of α—is found in the On-Line Encyclopedia of Integer Sequences [https://oeis.org/ol.html] as A004523 [“two even followed by one odd”] and as A232007 [“maximal number of moves needed to reach every square by a knight from a fixed position on an n×n chessboard, or −1 if it is not possible to reach every square”].)For the complementary set of four upper parameters of the7F6 function, the entries in order of increasing magnitude are expressible as(13)u3k=α+153k-45+2k-35+2k-25+3k-15+16,u4k=α+153k-45+2k-35+k-25+4k-15+17,u5k=α+152k-45+3k-35+k-25+4k-15+18,u6k=α+152k-45+3k-35+k-25+4k-15+19.For k=1, for illustrative purposes, application of these four rules yields α+9/5,α+11/5,α+12/5,α+13/5, and for k=5, we have α+16/5,α+17/5,α+18/5,α+19/5. For arbitrary k, the sum of the four terms under discussion minus 4α is an integer, namely, 2k-4/5+2k-3/5+(k-2)/5+3k-1/5+14. Further, let us note that, for integral values of k, u3k has values (14)1250502k+7+50-105-3sin⁡25π1-2k+2sin⁡4πk5-2sin⁡6πk5-3sin⁡15π-6πk+2sin⁡15π-4πk-3sin⁡253πk+π+sin⁡154πk+π+2sin⁡156πk+π+105+5-2sin⁡25π1-4k-2sin⁡2πk5+2sin⁡8πk5-2sin⁡25πk+1-3sin⁡15π-2πk-sin⁡254πk+π+sin⁡158πk+π+3cos⁡1104πk+π.


## 3.1.1. The Six Lower Parameters

The lower (bottom) six parametersbik, i=1,…,6, of the 7F6 function conform for all eleven cases to the simple linear rule(9)b1k,b2k,b3k,b4k,b5k,b6k=α+2k5+2310,α+2k5+52,α+2k5+2710,α+2k5+2910,α+2k5+3110,α+k+3.The six entries are summed to 6α+3k+33/2.


## 3.1.2. The Six Upper Parameters

The six upper parameters (aside from the seventhk-invariant constant of 2 already indicated), {u1k,u2k,u3k,u4k,u5k,u6k}, can be broken into one set of two (the numerical parts summing to integers), incorporating consecutive fractions having 6’s in their denominators, and one set of four (the numerical parts also summed to integers), incorporating consecutive fractions having 5’s in their denominators. For the set of two, the smaller of the two upper entries abides by the rule(10)u1k=α+164k3+2k+13+11,where the (integer-valued) floor function is employed, while the larger entry is given by(11)u2k=α+162k3+4k+13+13.For integral values of k, the same values of ui1 and ui2 are yielded by the interpolating functions(12)α+1186k+2cos⁡2πk3+2cos⁡4πk3+29,α+1186k-3sin⁡2πk3+3sin⁡4πk3+cos⁡2πk3+cos⁡4πk3+37,respectively.Fork=1, for illustrative purposes, application of the two rules yields α+11/6,α+13/6, and for k=5, we have α+19/6,α+23/6. (We have noted that u1k+u2k-2α=k/3+k+1/3+4 is an integer. The sequence of these integers—for arbitrary integer or half-integer values of α—is found in the On-Line Encyclopedia of Integer Sequences [https://oeis.org/ol.html] as A004523 [“two even followed by one odd”] and as A232007 [“maximal number of moves needed to reach every square by a knight from a fixed position on an n×n chessboard, or −1 if it is not possible to reach every square”].)For the complementary set of four upper parameters of the7F6 function, the entries in order of increasing magnitude are expressible as(13)u3k=α+153k-45+2k-35+2k-25+3k-15+16,u4k=α+153k-45+2k-35+k-25+4k-15+17,u5k=α+152k-45+3k-35+k-25+4k-15+18,u6k=α+152k-45+3k-35+k-25+4k-15+19.For k=1, for illustrative purposes, application of these four rules yields α+9/5,α+11/5,α+12/5,α+13/5, and for k=5, we have α+16/5,α+17/5,α+18/5,α+19/5. For arbitrary k, the sum of the four terms under discussion minus 4α is an integer, namely, 2k-4/5+2k-3/5+(k-2)/5+3k-1/5+14. Further, let us note that, for integral values of k, u3k has values (14)1250502k+7+50-105-3sin⁡25π1-2k+2sin⁡4πk5-2sin⁡6πk5-3sin⁡15π-6πk+2sin⁡15π-4πk-3sin⁡253πk+π+sin⁡154πk+π+2sin⁡156πk+π+105+5-2sin⁡25π1-4k-2sin⁡2πk5+2sin⁡8πk5-2sin⁡25πk+1-3sin⁡15π-2πk-sin⁡254πk+π+sin⁡158πk+π+3cos⁡1104πk+π.


## 3.2. DistinguishedF76 Function with 1 as an Upper Parameter in Q(k,α)

Eachk-specific formula Q(k,α) we have found also incorporates a second F76 function (again with argument z=27/64, which is, to repeat, invariably the case throughout this paper), having all its thirteen parameters simply equalling 1 less than those in the function just described. (A basic transformation exists [consulting the HYP manual of C. Krattenthaler, available at http://mathematik.univie.ac.at], allowing one to convert the thirteen [twelve α-dependent parameters, plus 1] of this F76 function [i.e., add 1 to each of them] to those thirteen of the first distinguished F76 previously described, plus other terms.)


## 3.3. Themk Remaining Fpp-1 Functions, p=8,…,8+mk-2, in Q(k,α)

Now, in addition to the two distinguishedF76 functions just presented, there are mk more hypergeometric functions Fpp-1, p>7, for each k, where(15)m-1,m0,m1,m2,m3,m4,m5,m6,m7,m8,m9=3,5,5,6,6,7,9,8,10,10,10.Each of these additional functions possesses, to begin with, the same seven upper parameters (that is, 2, plus those six indicated in (10), (11), and (13)) and the same six lower parameters (9), as in the first 7F6 function detailed above (Section 3.1). Then, the seven upper parameters are supplemented by from 1 to mk 2’s, and the six lower parameters supplemented by 1 to mk 1’s.


## 3.4. Largeα-Free Terms Collapsing to 0

We now point out a rather remarkable property of the formulas forQ(k,α) yielded by the FindSequenceFunction command. If we isolate those (often quite bulky) terms that do not involve any of the mk+2 hypergeometric functions for each k already described above, we find (to hundreds of digits of accuracy) that they collapse to zero. These terms, typically, do contain hypergeometric functions similar in nature to those described above, but with the crucial difference that the Dyson-index-like parameter α doesnot occur among their upper and lower parameters. Thus, we are left, after this nullification of terms, with formulas Q(k,α) that are simply sums of mk+2 polynomial-weighted Fpp-1 functions (of α), with p=7,7,8,…,7+mk.


## 3.5. Summary

To reiterate, for eachk, our formulas for Q(k,α) all contain a single function of the form(16)F762,uk,u2k,u3k,u4k,u5k,u6k;bk,b2k,b3k,b4k,b5k,b6k;2764.There is another distinguished single F76 function, with all its thirteen parameters being one less. Also there are mk additional functions, i=1,…,mk,(17)F7+i6+i2,2,…,uk,u2k,u3k,u4k,u5k,u6k;1,…,bk,b2k,b3k,b4k,b5k,b6k;2764,with the number of upper 2’s running from 2 to mk+1 and the number of lower 1’s, simultaneously running from 1 to mk.


## 4. Decomposition ofQ(k,α) into the Product G1k(α)G2k(α)

The formulas forQ(k,α) that we have obtained can all be written—we have found—in the product form G1k(α)G2k(α). The G2k(α) factor involves the summation of the hypergeometric functions Fpp-1 indicated above, each such function weighted by a polynomial in α, the degrees of the weighting polynomials diminishing as p increases. Let us first discuss the other (hypergeometric-free) factor G1k(α), involving ratios of products of Pochhammer symbols.


### 4.1. Hypergeometric-Function-Independent FactorG1k(α)

Some supplementary computations (involving an independent use of the FindSequenceFunction command) indicated that these (hypergeometric-free) factors can be written quite concisely, in terms of the upper and lower parameter sets, settingUik=uik-α, Bik=bik+1-α, as(18)G1kα=2764α-1U1kα-1U2kα-1U3kα-1U4kα-1U5kα-1U6kα-1B1kα-1B2kα-1B3kα-1B4kα-1B5kα-1B6kα-1,where the Pochhammer symbol (rising factorial) is employed. We note that, remarkably, G1k(1)=1, further apparent indication of the special/privileged status of the standard (complex, α=1) two-qubit states.


### 4.2. Hypergeometric-Function-Dependent FactorG1k(α)

#### 4.2.1. Canonical Form

In AppendixA, for k=-1,0,1,2, we show the “canonical form” we have developed for the factors G2k(α) (cf. [17, Fig. 3]); the component hypergeometric parts of which we have discussed in Section 3.


## 4.1. Hypergeometric-Function-Independent FactorG1k(α)

Some supplementary computations (involving an independent use of the FindSequenceFunction command) indicated that these (hypergeometric-free) factors can be written quite concisely, in terms of the upper and lower parameter sets, settingUik=uik-α, Bik=bik+1-α, as(18)G1kα=2764α-1U1kα-1U2kα-1U3kα-1U4kα-1U5kα-1U6kα-1B1kα-1B2kα-1B3kα-1B4kα-1B5kα-1B6kα-1,where the Pochhammer symbol (rising factorial) is employed. We note that, remarkably, G1k(1)=1, further apparent indication of the special/privileged status of the standard (complex, α=1) two-qubit states.


## 4.2. Hypergeometric-Function-Dependent FactorG1k(α)

### 4.2.1. Canonical Form

In AppendixA, for k=-1,0,1,2, we show the “canonical form” we have developed for the factors G2k(α) (cf. [17, Fig. 3]); the component hypergeometric parts of which we have discussed in Section 3.


## 4.2.1. Canonical Form

In AppendixA, for k=-1,0,1,2, we show the “canonical form” we have developed for the factors G2k(α) (cf. [17, Fig. 3]); the component hypergeometric parts of which we have discussed in Section 3.


## 5. Difference Equation Formulas forG2k(α)

It further appears that all theG2k(α) factors (k=-1,0,1,…,9) (Appendix B) can be equivalently written as functions that satisfy first-order difference (recurrence) equations of the form(19)p0kα+p1kαG2kα+p2kαG2k1+α=0,where p’s are polynomials in α (cf. [29]). This finding was established by yet another application of the Mathematica FindSequenceFunction command.That is, we generated—for each value ofk under consideration—a sequence (α=1,2,…,85) of the rational values yielded by the hypergeometric-based formulas for G2k(α), to which the command was then applied. While we have limited ourselves in Appendix B to displaying our results for k=-1,0,1,2,3, and 4, we do have the analogous set of results in terms of the hypergeometric functions for the additional instances, k=5,6,7,8, and 9 and presume that an equivalent set of difference equation results is constructible (though substantial efforts with k=5 have not to this point succeeded). The initial points G2k(1) in the six difference equations shown are—in the indicated order—1/14,4/33,45/286,1553/8398,3073/14858,8348/37145. The next five members of this monotonically increasing sequence are {188373/785726,1096583/4342170, 6050627/22951470,160298199/586426690, 13988600951/49611697974}. Since, as noted above, G1k(1)=1, these are the respective separability probabilities Q(k,1) themselves. We would like to extend this sequence sufficiently, so that we might be able to establish an underlying rule for it. (However, since the sequence is increasing in value, the Legendre-polynomial density-approximation procedure of Provost converges more slowly as α increases, so our quest seems somewhat problematical, despite the large number [15,801] of moments incorporated [cf. [1, App. II]].)If in the difference equation fork=-1, we replace the term G2-1(1)=1/14 by G2-1(1)=0, then we can add(20)π3-3α-543α+255α+39/10α+111/10α+113/10α+13/2α+117/10α+1ΓαΓα+252055003Γ5αΓα+1/6Γα+5/6to the α-specific values obtained from the so-modified equation to recover the values generated by the original k=-1 difference equation.


### 5.1. Polynomial Coefficients in Difference Equations

#### 5.1.1. The Polynomialsp2k(α)

We have for the six (k=-1,0,1,2,3,4) cases at hand (Appendix B) the proportionality relation(21)p2kα∝∏i=16uik-1,where uik’s (and bik’s)—as indicated in Section 3—are themselves functions of α.


#### 5.1.2. The Polynomialsp1k(α)

For all six displayed cases,(22)p1kα∝∏i=16bik.


#### 5.1.3. The Polynomialsp0k(α)

Further, for all six cases, the polynomial coefficientsp0k(α)—constituting theinhomogeneous parts of the recurrences—are proportional to the product of a factor of the form(23)∏i=16bikbik-1and an irreducible polynomial. These irreducible polynomials are, in the indicated order (k=-1,0,1),(24)9250α4+12625α3+5645α2+938α+54,(25)185000α5+779750α4+1289125α3+1042015α2+410694α+63000,(26)74000α6+578300α5+1830820α4+3013197α3+2724024α2+1284280α+246960,and (for k=2)(27)740000α7+9002000α6+45576950α5+125164535α4+202090226α3+192332891α2+100092606α+22004136.The irreducible polynomial for k=3 is also of degree 7; that is,(28)740000α7+11666000α6+76382750α5+271168745α4+566336789α3+698007782α2+471120306α+134548128.For k=4, this auxiliary polynomial p0k(α) is now the product of (9+4α) times an irreducible polynomial of degree 7; that is,(29)296000α7+5584000α6+43492140α5+182972656α4+451645197α3+656629192α2+522054355α+175452420.The coefficients of the highest powers of α in all six irreducible polynomials are factorable into the product of 37 and powers of 2 and 5.


## 5.1. Polynomial Coefficients in Difference Equations

### 5.1.1. The Polynomialsp2k(α)

We have for the six (k=-1,0,1,2,3,4) cases at hand (Appendix B) the proportionality relation(21)p2kα∝∏i=16uik-1,where uik’s (and bik’s)—as indicated in Section 3—are themselves functions of α.


### 5.1.2. The Polynomialsp1k(α)

For all six displayed cases,(22)p1kα∝∏i=16bik.


### 5.1.3. The Polynomialsp0k(α)

Further, for all six cases, the polynomial coefficientsp0k(α)—constituting theinhomogeneous parts of the recurrences—are proportional to the product of a factor of the form(23)∏i=16bikbik-1and an irreducible polynomial. These irreducible polynomials are, in the indicated order (k=-1,0,1),(24)9250α4+12625α3+5645α2+938α+54,(25)185000α5+779750α4+1289125α3+1042015α2+410694α+63000,(26)74000α6+578300α5+1830820α4+3013197α3+2724024α2+1284280α+246960,and (for k=2)(27)740000α7+9002000α6+45576950α5+125164535α4+202090226α3+192332891α2+100092606α+22004136.The irreducible polynomial for k=3 is also of degree 7; that is,(28)740000α7+11666000α6+76382750α5+271168745α4+566336789α3+698007782α2+471120306α+134548128.For k=4, this auxiliary polynomial p0k(α) is now the product of (9+4α) times an irreducible polynomial of degree 7; that is,(29)296000α7+5584000α6+43492140α5+182972656α4+451645197α3+656629192α2+522054355α+175452420.The coefficients of the highest powers of α in all six irreducible polynomials are factorable into the product of 37 and powers of 2 and 5.


## 5.1.1. The Polynomialsp2k(α)

We have for the six (k=-1,0,1,2,3,4) cases at hand (Appendix B) the proportionality relation(21)p2kα∝∏i=16uik-1,where uik’s (and bik’s)—as indicated in Section 3—are themselves functions of α.


## 5.1.2. The Polynomialsp1k(α)

For all six displayed cases,(22)p1kα∝∏i=16bik.


## 5.1.3. The Polynomialsp0k(α)

Further, for all six cases, the polynomial coefficientsp0k(α)—constituting theinhomogeneous parts of the recurrences—are proportional to the product of a factor of the form(23)∏i=16bikbik-1and an irreducible polynomial. These irreducible polynomials are, in the indicated order (k=-1,0,1),(24)9250α4+12625α3+5645α2+938α+54,(25)185000α5+779750α4+1289125α3+1042015α2+410694α+63000,(26)74000α6+578300α5+1830820α4+3013197α3+2724024α2+1284280α+246960,and (for k=2)(27)740000α7+9002000α6+45576950α5+125164535α4+202090226α3+192332891α2+100092606α+22004136.The irreducible polynomial for k=3 is also of degree 7; that is,(28)740000α7+11666000α6+76382750α5+271168745α4+566336789α3+698007782α2+471120306α+134548128.For k=4, this auxiliary polynomial p0k(α) is now the product of (9+4α) times an irreducible polynomial of degree 7; that is,(29)296000α7+5584000α6+43492140α5+182972656α4+451645197α3+656629192α2+522054355α+175452420.The coefficients of the highest powers of α in all six irreducible polynomials are factorable into the product of 37 and powers of 2 and 5.


## 6. Hypergeometric-Free Formulas forQ(k+1,α)-Q(k,α)

In AppendixC we show formulas we have generated for the differences between the formulas for Q(k,α) for successive values of k. We note that these are hypergeometric-free. We will find below (64) that these obey the formula(30)Qk+1,α-Qk,α=π3-3α-1αΓ3α+3/220α+8k+11Γk+2α+3/2Γk+3α+3/2Γ2k+5α+22Γα+1/2Γα+5/6Γα+7/6Γk+α+2Γk+4α+2Γ2k+5α+7/2.


## 7. Partial Separability Probability Asymptotics

### 7.1.k-Specific prob(ρPT>ρ) Formulas

Now, as concerns the eleven formulasQ(k,α) (k=-1,0,1,…,9) we have obtained for prob(ρPT>ρ), which have been the principal focus of the paper, we have computed the ratios of the probability for α=101 to the probability for α=100. These ranged from 0.419810 (k=-1) to 0.4204296 (k=9). Let us note here that z=27/64≈0.421875.


### 7.2.α-Specific prob(ρPT>ρ) Formulas

We had availableα=1/2,1, and 2 computations for k=1,…,40 for this scenario. We found that, for each of the three values of α, we could construct strongly linear plots—with unit-like slopes between 1.00177 and 1.00297—by taking k times the ratio (R) of the (k+1) separability probability to the kth separability probability. (From this, it appears, simply, that R→1, as k→∞.)


### 7.3. “Diagonal”α=kprob(ρPT>ρ) Formulas

For valuesα=k=1,…,50, we were able to construct a strongly linear plot by—similarly to the immediate last analysis—taking k=α times the ratio of the (k+1)=(α+1) separability probability to the k=αth separability probability. Now, however, rather than a slope very close to 1, we found a slope near to one-half, that is, 0.486882. The (k=α=0)-intercept of the estimated line was 0.894491.


## 7.1.k-Specific prob(ρPT>ρ) Formulas

Now, as concerns the eleven formulasQ(k,α) (k=-1,0,1,…,9) we have obtained for prob(ρPT>ρ), which have been the principal focus of the paper, we have computed the ratios of the probability for α=101 to the probability for α=100. These ranged from 0.419810 (k=-1) to 0.4204296 (k=9). Let us note here that z=27/64≈0.421875.


## 7.2.α-Specific prob(ρPT>ρ) Formulas

We had availableα=1/2,1, and 2 computations for k=1,…,40 for this scenario. We found that, for each of the three values of α, we could construct strongly linear plots—with unit-like slopes between 1.00177 and 1.00297—by taking k times the ratio (R) of the (k+1) separability probability to the kth separability probability. (From this, it appears, simply, that R→1, as k→∞.)


## 7.3. “Diagonal”α=kprob(ρPT>ρ) Formulas

For valuesα=k=1,…,50, we were able to construct a strongly linear plot by—similarly to the immediate last analysis—taking k=α times the ratio of the (k+1)=(α+1) separability probability to the k=αth separability probability. Now, however, rather than a slope very close to 1, we found a slope near to one-half, that is, 0.486882. The (k=α=0)-intercept of the estimated line was 0.894491.


## 8. Total Separability Probability Formulas

Efforts of our to conduct parallel sets of (k-specific) analyses to those reported above for thetotal separability probabilities P(k,α), corresponding to ρPT>0, rather than for that component part Q(k,α) of the probabilities satisfying the determinantal inequality ρPT>ρ had been unsuccessful, in the following sense. We had computed what appeared to be appropriate sequences (α=1,2,…,74) of rational values for k=1 and (α=1,2,…,124) for k=2, but the Mathematica FindSequenceFunction did not yield any underlying governing rules. (This can be contrasted with the results in [1], where such successes were reported in obtaining α-specific [ρPT>0] formulas [α=1,2,…,13 and 1/2,3/2,5/2,7/2], including (4)–(6). However, we do eventually succeed in characterizing the nature of these two (k=1,2) sequences [cf. Appendix G].)In Figure3, we plot the logs of these k=1 seventy-four total separability probabilities (based on α=1,…,74). A least-squares linear fit to these points is -0.878482α-0.362781, while in Figure 4, we show (based on α=1,…,124) the k=2 counterpart, with an analogous fit of -0.871033α+0.351201. (We note that log⁡(27/64)≈-0.863046.) Although the slopes of these two linear fits are quite close, the y-intercepts themselves are of different sign. The predicted probabilities at α=1, the first of the fitted points, are 0.289019 and 0.602955, respectively. In statistical parlance, the “coefficients of determination” or R2 for the two linear fits to the log-plots are both greater than 0.99995. Further, sampling at α=1,51,101,…,1451, we obtained an estimated, again, very well-fitting line of -1.4754-0.86417α.Figure 3
Plot of logs of total separability probability (ρPT>0) for random induced measure with k=1. A least-squares linear fit to these 74 points is -0.878482α-0.362781.Figure 4
Plot of logs of total separability probability (ρPT>0) for random induced measure with k=2. A least-squares linear fit to these 124 points is -0.871033α+0.351201. We note that log⁡(27/64)≈-0.863046.


### 8.1. Total Separability Probability Asymptotics

#### 8.1.1.k-Specific prob(ρPT>0) Formulas

Dunkl, on the basis of ourk=1, α=1,51,101,…,1451 analysis just above (and its companions), did advance the bold and (certainly, in our overall analytical context) elegant hypothesis of a k-invariant (α→∞) slope equal to log⁡(27/64)≈-0.8630462173553, which does seem quite consistent with the numerical properties we have observed (i.e., with the direction in which the estimates of the slope tend as the number of points sampled increase).As further support, we obtained for ak=2, α=1,49,73,…,1465 analysis, a slope estimate of -0.864025, again converging in the direction of log⁡27/64. (Let us remark, regarding the generalized two-qubit version of the [simpler, lower-dimensional]X-states model [20, 30, 31], that it has been shown that the slope of a [now, log-log] plot of log⁡(probρPT>0) versus log⁡α tends to -1/2, as α→∞.)


#### 8.1.2.α-Specific prob(ρPT>0) Formulas

These interesting observations led us to reexamine, for their asymptotic properties, the “dual”P(k,α) formulas (4)–(6), given above and previously reported in [1]. We now find—through analytic means—that for each of α=1,2,3,4, and 1/2,3/2,5/2,9/2, that, as k→∞, the ratio of the logarithm of the (k+1)-st separability probability to the logarithm of the kth separability probability is 16/27 (cf. [32, sec. 7]). (Presumably, the pattern continues for larger α, but the required computations have, so far, proved too challenging.)For example, forα=1/2, we have for the two-rebit total separability probability P(k,1/2), as a function of k, formula (6) given above. In Figure 5, we show a plot of log⁡-log⁡Pk,1/2 versus k. The slope of a least-squares-fitted line based on the 200 points is -0.523280, while log⁡16/27⁡≈-0.523248. (As we increase α from 1/2 but hold the number of points constant at 200, the approximation of the slope to this value slowly weakens.)Figure 5
Plot oflog⁡(-log⁡Pk,1/2) versus k. The slope of a least-squares-fitted line is −0.523280, while log⁡16/27≈-0.523248.


## 8.1. Total Separability Probability Asymptotics

### 8.1.1.k-Specific prob(ρPT>0) Formulas

Dunkl, on the basis of ourk=1, α=1,51,101,…,1451 analysis just above (and its companions), did advance the bold and (certainly, in our overall analytical context) elegant hypothesis of a k-invariant (α→∞) slope equal to log⁡(27/64)≈-0.8630462173553, which does seem quite consistent with the numerical properties we have observed (i.e., with the direction in which the estimates of the slope tend as the number of points sampled increase).As further support, we obtained for ak=2, α=1,49,73,…,1465 analysis, a slope estimate of -0.864025, again converging in the direction of log⁡27/64. (Let us remark, regarding the generalized two-qubit version of the [simpler, lower-dimensional]X-states model [20, 30, 31], that it has been shown that the slope of a [now, log-log] plot of log⁡(probρPT>0) versus log⁡α tends to -1/2, as α→∞.)


### 8.1.2.α-Specific prob(ρPT>0) Formulas

These interesting observations led us to reexamine, for their asymptotic properties, the “dual”P(k,α) formulas (4)–(6), given above and previously reported in [1]. We now find—through analytic means—that for each of α=1,2,3,4, and 1/2,3/2,5/2,9/2, that, as k→∞, the ratio of the logarithm of the (k+1)-st separability probability to the logarithm of the kth separability probability is 16/27 (cf. [32, sec. 7]). (Presumably, the pattern continues for larger α, but the required computations have, so far, proved too challenging.)For example, forα=1/2, we have for the two-rebit total separability probability P(k,1/2), as a function of k, formula (6) given above. In Figure 5, we show a plot of log⁡-log⁡Pk,1/2 versus k. The slope of a least-squares-fitted line based on the 200 points is -0.523280, while log⁡16/27⁡≈-0.523248. (As we increase α from 1/2 but hold the number of points constant at 200, the approximation of the slope to this value slowly weakens.)Figure 5
Plot oflog⁡(-log⁡Pk,1/2) versus k. The slope of a least-squares-fitted line is −0.523280, while log⁡16/27≈-0.523248.


## 8.1.1.k-Specific prob(ρPT>0) Formulas

Dunkl, on the basis of ourk=1, α=1,51,101,…,1451 analysis just above (and its companions), did advance the bold and (certainly, in our overall analytical context) elegant hypothesis of a k-invariant (α→∞) slope equal to log⁡(27/64)≈-0.8630462173553, which does seem quite consistent with the numerical properties we have observed (i.e., with the direction in which the estimates of the slope tend as the number of points sampled increase).As further support, we obtained for ak=2, α=1,49,73,…,1465 analysis, a slope estimate of -0.864025, again converging in the direction of log⁡27/64. (Let us remark, regarding the generalized two-qubit version of the [simpler, lower-dimensional]X-states model [20, 30, 31], that it has been shown that the slope of a [now, log-log] plot of log⁡(probρPT>0) versus log⁡α tends to -1/2, as α→∞.)


## 8.1.2.α-Specific prob(ρPT>0) Formulas

These interesting observations led us to reexamine, for their asymptotic properties, the “dual”P(k,α) formulas (4)–(6), given above and previously reported in [1]. We now find—through analytic means—that for each of α=1,2,3,4, and 1/2,3/2,5/2,9/2, that, as k→∞, the ratio of the logarithm of the (k+1)-st separability probability to the logarithm of the kth separability probability is 16/27 (cf. [32, sec. 7]). (Presumably, the pattern continues for larger α, but the required computations have, so far, proved too challenging.)For example, forα=1/2, we have for the two-rebit total separability probability P(k,1/2), as a function of k, formula (6) given above. In Figure 5, we show a plot of log⁡-log⁡Pk,1/2 versus k. The slope of a least-squares-fitted line based on the 200 points is -0.523280, while log⁡16/27⁡≈-0.523248. (As we increase α from 1/2 but hold the number of points constant at 200, the approximation of the slope to this value slowly weakens.)Figure 5
Plot oflog⁡(-log⁡Pk,1/2) versus k. The slope of a least-squares-fitted line is −0.523280, while log⁡16/27≈-0.523248.


## 9. “Concise Formulas” forQ(k,α)

Let us remind the reader of the interesting “concise” (Hilbert-Schmidt [k=0]) generalized two-qubit result—applying Zeilberger’s (“creative telescoping”) algorithm [33]—of Qing-Hu Hou, reported in [17, eqs. (1)–(3)]. This—in our present notation—takes the form(31)Q0,α=∑i=0∞f0α+i,where(32)f0α=Q0,α-Q0,α+1=q0α2-4α-6Γ3α+5/2Γ5α+26Γα+1Γ2α+3Γ5α+13/2,(33)q0α=185000α5+779750α4+1289125α3+1042015α2+410694α+63000=α5α25α2α740α+3119+10313+208403+410694+63000.We divide the originally reported formula by one-half [9], since we have moved here from the (k=0) Hilbert-Schmidt ρPT>0 original scenario to its ρPT>ρ counterpart. Using our earlier results above, Hou has further been able to construct the k=1 analogue of the “concise formula” above (a Maple worksheet of his is presented in Appendix D [cf. [17, Figs. 5, 6]). That is,(34)Q1,α=∑i=0∞f1α+i,where(35)f1α=q1α27αΓ5αΓα+5/6Γα+7/650000αΓαΓα+17/10Γα+19/10Γα+21/10Γα+23/10Γ2α+5,(36)q1α=9π10000005α+15α+25α+3×74000α6+578300α5+1830820α4+3013197α3+2724024α2+1284280α+246960.(These results correspond to the variable “dif” in Appendix D.) Thus, in passing from the (symmetric k=0) Hilbert-Schmidt setting to the random induced k=1 scenario, the degree of “conciseness” somewhat diminishes. The polynomials q0(α) and q1(α) in this pair of formulas are the same as the difference equation (19) polynomials p00(α) and p01(α), given in (25) and (26).At this point in our research, we were able to employ the Mathematica-based HolonomicFunctions package of Christoph Koutschan of the Research Institute for Symbolic Computation (RISC) of Johannes Kepler University. With it, we were readily able to derive thek=-1 result(37)Q-1,α=P-1,α=∑i=0∞f-1α+i,where(38)f-1α=π5-5α-416-α-127αα10α+7925α2+615α+134+54Γα+1/6Γα+5/6Γ5α+1Γα+9/10Γα+1Γα+11/10Γα+13/10Γα+17/10Γ2α+2.We see that the polynomial α(10α+7)(925α2+615α+134)+54 above is, in expanded form, the same as p0-1(α) given in (24).For the standard trio of Dyson indicesα=1/2,1, and 2, this formula for Q(-1,α) yields 1/8,1/14, and 11/442, respectively, while α=-1/2,0 lead to 1/3,1/5. (Also, α=-3/2 gives 1/3, and α=-1 yields 1/5.) Additionally, α=-1/3 gives 19/60C2, where C2 is Baxter’s four-coloring constant for a triangular lattice; that is, C2=(3/4π2)Γ(1/3)3. (Also, α=2/3 gives 1-27C2/44.) Continuing with this “zoo” of remarkable results (suggested largely by use of WolframAlpha), α=1/4 gives 1-GGA≈0.1653731583, where GGA is Gauss’s constant, that is, the reciprocal of thearithmetic-geometric mean of 1 and 2, equalling Γ(1/4)2/22π3/2. Now, for α=-1/4, we get 8/5L+1≈1.6102078108, where L is the Lemniscate constant; that is, L=(1/22π)Γ(1/4)2. To continue, α=-2/3 gives us 1-163/1008I(ω1)≈0.87795554, where ω1=(1+i3)Γ(1/3)3/8π is a known constant of interest (cf. [17, sec. 3.2.1]).Further, employing the RISC package, we obtained(39)Q3,α=∑i=0∞f3α+i,where(40)f3α=33α+44-2α-52α+5Γα+8/5Γα+9/5Γα+11/6Γα+13/6Γα+11/5Γα+12/5q3α6255πα+4Γα+27/10Γα+29/10Γα+31/10Γα+33/10Γ2α+7,q3α=ααα5α50α8α370α+5833+305531+54233749+566336789+698007782+471120306+134548128is a degree 7 polynomial in α.For the standard trio of Dyson indicesα=1/2,1, and 2, this formula for Q(3,α) yields 84883/262144,3073/14858 and 3439/41354, respectively.It would clearly be of interest to find such “concise” expressions forQ(k,α), encompassing the four (k=-1,0,1,3) examples above, as well as values k>3. (We have so far encountered certain difficulties in applying the RISC HolonomicFunctions program to the k=2 scenario.)


## 10. Series of Exactk-Values for Certain α and Associated Formulas

### 10.1. Series

We have previously notedQ(-1,-1/3)=(19/60)C2, where C2 is Baxter’s four-coloring constant for a triangular lattice; that is, C2=(3/4π2)Γ(1/3)3. For the succeeding values k=0,…,9, we obtain C2/2+1,1-3C2/20, 1-783C2/3740, 1-1171341C2/4989160, 1-51068151C2/204555560, 1-132326834139C2/509547899960, 1-8028455705181C2/30063326097640, 1-582160729281381C2/2134496152932440, 1-4372426421400790827C2/15767523081711934280, 1-447620586926496661827C2/1592519831252905362280.For the series (k=-1,0,…,9) with α=-1/2, we obtain 1/3,1/3,1,13/16,191/256,1453/2048,44923/65536, 350323/52428,5494379/8388608,43249277/67108864, 2730885203/4294967296. Here, all the denominators (k=1,…,9) are simply increasing powers of 2.For the series (k=0,…,9) with α=-1/4, we obtain {1,1-4/5L,1-184/195L,1-1116/1105L, 1-504688/480675L,1-19161148/17784975L, 1-47082376/42893175L,1-301219589404/270527254725L, 1-18584575275424/16502162538225L, 1-288596382356/253879423665L}, where L is the indicated Lemniscate constant; that is, L=(1/22π)Γ(1/4)2.


### 10.2. Formulas

This last series has the explanatory rule (k=0,1,…,9)(41)Qk,-14=Γ5/4Γ2k+3/4F321,k+3/8,k+7/8;k+9/8,k+13/8;1/Γ2k+9/4-πLΓ3/4+1=2-2k-9/4Γ2k+34F~321,k+38,k+78;k+98,k+138;1+12,where theregularized hypergeometric function is indicated. For k=-1, the formula yields 1+4/5L, while our prior computations indicate a value of 1+8/5L.Also (now agreeing fork=-1,0,…,9),(42)Qk,14=1+L21πU,where(43)U=4F3258,1,98;118,158;1-21-4Γ11/4Γ2k+13/4F321,k+13/8,k+17/8;k+19/8,k+23/8;1Γ5/4Γ2k+19/4. Absorbing the Lemniscate constant L, we obtain, equivalently,(44)Qk,14=-2-2k-19/4Γ2k+134F~321,k+138,k+178;k+198,k+238;1+12. We see some obvious parallels between the formulas for Q(k,-1/4) and Q(k,1/4). (We note that Q(0,1/4)-Q(k,-1/4)=-17GGa/21, where Gauss’s constant is indicated.)In fact, we can subsume both these last two formulas (α=-1/4,1/4) into(45)Qk,α=12-2-5α-2k-7/2sgn⁡αΓ2k+5α+2F~321,k+5α2+1,k+5α2+32;k+5α2+74,k+5α2+94;1.Building upon (45), we found(46)Qk,34=-2-2k-29/4Γ2k+234F~321,k+238,k+278;k+298,k+338;1-Γ2k+23/42πk+3Γ2k+21/4+12.Strikingly simply, we have the result (valid for all eleven valuesk=-1,0,…,9 for which we have computations)(47)Qk,12=12-Γ2k+9/2πΓ2k+5,having a root at k=-3/2. So, using formula (6) above, we find that thecomplementary separability probability, that is, associated with the determinantal inequality ρ>ρPT≥0, is(48)Pk,12-Qk,12=Γ2k+9/21/Γ2k+5-4k+18k+15Γk+2/Γ3k+7π+12. Also, we have found (agreeing with the earlier formulas for all eleven k) that(49)Qk,-12=Γ2k-1/2πΓ2k+12, for k=1,2,…,9, with the results for k=-1,0 of 1/2 differing from the prediction of 1/3 given by the early formulas given above.Further, we have(50)Qk,1=12-4k+3Γk+7/22Γk+9/2πΓk+5Γ2k+13/2, having a root at k=-2.To continue (with a root atk=-5/2),(51)Qk,32=12-6k+31Γ2k+19/24πk+5k+6Γ2k+9.Further,(52)Qk,2=12-4k+6k+6Γk+11/2Γk+13/2Γk+15/2πΓk+9Γ2k+23/2,having a root at k=-3, agreeing with our earlier formulas for all eleven k (as well as k=-2 and 10).Our formulas give thatQ(k,0) is equal to 1/5 for both k=-2 and 1 and equal to 1/2 for k=0,…,9. Here, α=0 presumably corresponds to a classical/nonquantum scenario.Dunkl has observed that, for integral values ofα, the arguments of the gamma functions in the numerators are of the form {2α+k+3/2,2α+α/2+k+3/2,3α+k+3/2} and in the denominators of the form {k+4n+1,2k+5n+3/2}. He further noted that the leading coefficient (highest power) in the polynomial takes the form α25α+2k+1kα+α-1/2-1. Also, the second leading coefficient (normalizing the leading coefficient of the polynomial to 1) follows the rule(53)c2=148190α2-174α-3-1α10α+3-55.Similarly, the so-normalized leading third coefficient takes the form(54)c3=30766α4-77260α3+23350α2-5-1α2α950α2-885α-716-213+26920α+37993840.We have been able to generate a considerable number (includingk=1,…,100) of such Q(k,α) formulas, a limited number of which we present in Appendix E.Each half-integralα formula contains a gamma function in its numerator with an argument of the form 2+5α+2k and in its denominator a gamma function with an argument of the form 2k+(1/2)-1α(-2(-1)α(5α+2)-i).


### 10.3. Sets of Consecutive Negative Roots

All theQ(k,α) formulas we have (Appendix E), for nonnegative half-integer and integer values of α, have roots (in unit steps) from k=-α-1 downwards to k=-(1/4)(-1)α((-1)α(10α+1)-1). So, there are(55)-α+14-1α-1α10α+1-1-1,associated roots. (The formulas displayed in Appendix E with negative values of α match our computations only above certain [nonnegative] values of k.)


## 10.1. Series

We have previously notedQ(-1,-1/3)=(19/60)C2, where C2 is Baxter’s four-coloring constant for a triangular lattice; that is, C2=(3/4π2)Γ(1/3)3. For the succeeding values k=0,…,9, we obtain C2/2+1,1-3C2/20, 1-783C2/3740, 1-1171341C2/4989160, 1-51068151C2/204555560, 1-132326834139C2/509547899960, 1-8028455705181C2/30063326097640, 1-582160729281381C2/2134496152932440, 1-4372426421400790827C2/15767523081711934280, 1-447620586926496661827C2/1592519831252905362280.For the series (k=-1,0,…,9) with α=-1/2, we obtain 1/3,1/3,1,13/16,191/256,1453/2048,44923/65536, 350323/52428,5494379/8388608,43249277/67108864, 2730885203/4294967296. Here, all the denominators (k=1,…,9) are simply increasing powers of 2.For the series (k=0,…,9) with α=-1/4, we obtain {1,1-4/5L,1-184/195L,1-1116/1105L, 1-504688/480675L,1-19161148/17784975L, 1-47082376/42893175L,1-301219589404/270527254725L, 1-18584575275424/16502162538225L, 1-288596382356/253879423665L}, where L is the indicated Lemniscate constant; that is, L=(1/22π)Γ(1/4)2.


## 10.2. Formulas

This last series has the explanatory rule (k=0,1,…,9)(41)Qk,-14=Γ5/4Γ2k+3/4F321,k+3/8,k+7/8;k+9/8,k+13/8;1/Γ2k+9/4-πLΓ3/4+1=2-2k-9/4Γ2k+34F~321,k+38,k+78;k+98,k+138;1+12,where theregularized hypergeometric function is indicated. For k=-1, the formula yields 1+4/5L, while our prior computations indicate a value of 1+8/5L.Also (now agreeing fork=-1,0,…,9),(42)Qk,14=1+L21πU,where(43)U=4F3258,1,98;118,158;1-21-4Γ11/4Γ2k+13/4F321,k+13/8,k+17/8;k+19/8,k+23/8;1Γ5/4Γ2k+19/4. Absorbing the Lemniscate constant L, we obtain, equivalently,(44)Qk,14=-2-2k-19/4Γ2k+134F~321,k+138,k+178;k+198,k+238;1+12. We see some obvious parallels between the formulas for Q(k,-1/4) and Q(k,1/4). (We note that Q(0,1/4)-Q(k,-1/4)=-17GGa/21, where Gauss’s constant is indicated.)In fact, we can subsume both these last two formulas (α=-1/4,1/4) into(45)Qk,α=12-2-5α-2k-7/2sgn⁡αΓ2k+5α+2F~321,k+5α2+1,k+5α2+32;k+5α2+74,k+5α2+94;1.Building upon (45), we found(46)Qk,34=-2-2k-29/4Γ2k+234F~321,k+238,k+278;k+298,k+338;1-Γ2k+23/42πk+3Γ2k+21/4+12.Strikingly simply, we have the result (valid for all eleven valuesk=-1,0,…,9 for which we have computations)(47)Qk,12=12-Γ2k+9/2πΓ2k+5,having a root at k=-3/2. So, using formula (6) above, we find that thecomplementary separability probability, that is, associated with the determinantal inequality ρ>ρPT≥0, is(48)Pk,12-Qk,12=Γ2k+9/21/Γ2k+5-4k+18k+15Γk+2/Γ3k+7π+12. Also, we have found (agreeing with the earlier formulas for all eleven k) that(49)Qk,-12=Γ2k-1/2πΓ2k+12, for k=1,2,…,9, with the results for k=-1,0 of 1/2 differing from the prediction of 1/3 given by the early formulas given above.Further, we have(50)Qk,1=12-4k+3Γk+7/22Γk+9/2πΓk+5Γ2k+13/2, having a root at k=-2.To continue (with a root atk=-5/2),(51)Qk,32=12-6k+31Γ2k+19/24πk+5k+6Γ2k+9.Further,(52)Qk,2=12-4k+6k+6Γk+11/2Γk+13/2Γk+15/2πΓk+9Γ2k+23/2,having a root at k=-3, agreeing with our earlier formulas for all eleven k (as well as k=-2 and 10).Our formulas give thatQ(k,0) is equal to 1/5 for both k=-2 and 1 and equal to 1/2 for k=0,…,9. Here, α=0 presumably corresponds to a classical/nonquantum scenario.Dunkl has observed that, for integral values ofα, the arguments of the gamma functions in the numerators are of the form {2α+k+3/2,2α+α/2+k+3/2,3α+k+3/2} and in the denominators of the form {k+4n+1,2k+5n+3/2}. He further noted that the leading coefficient (highest power) in the polynomial takes the form α25α+2k+1kα+α-1/2-1. Also, the second leading coefficient (normalizing the leading coefficient of the polynomial to 1) follows the rule(53)c2=148190α2-174α-3-1α10α+3-55.Similarly, the so-normalized leading third coefficient takes the form(54)c3=30766α4-77260α3+23350α2-5-1α2α950α2-885α-716-213+26920α+37993840.We have been able to generate a considerable number (includingk=1,…,100) of such Q(k,α) formulas, a limited number of which we present in Appendix E.Each half-integralα formula contains a gamma function in its numerator with an argument of the form 2+5α+2k and in its denominator a gamma function with an argument of the form 2k+(1/2)-1α(-2(-1)α(5α+2)-i).


## 10.3. Sets of Consecutive Negative Roots

All theQ(k,α) formulas we have (Appendix E), for nonnegative half-integer and integer values of α, have roots (in unit steps) from k=-α-1 downwards to k=-(1/4)(-1)α((-1)α(10α+1)-1). So, there are(55)-α+14-1α-1α10α+1-1-1,associated roots. (The formulas displayed in Appendix E with negative values of α match our computations only above certain [nonnegative] values of k.)


## 11. Hypergeometric Formula forQ(k,α)

Based on the information presented above, including that, in an extended form of AppendixE, Dunkl developed the following formula, succeeding in reproducing our computations for α=0,1,2,…:(56)Qk,α=Q-α,α∑j=0α+kHα,j,where(57)Q-α,α=12427α3/4α5/4α5/6α7/6α,(58)Hα,j=3α/2jα+1/2j3α/2+1/2j3α/2+11/8j2α+1/2jj!3α/2+3/8j3α/2+3/4j3α/2+5/4j3α+1j.(In explaining how this formula was obtained, Dunkl stated that the key insights were that Q(k+1,α)-Q(k,α) factors nicely and Q(-α-1,α)=0.) If we let both α and k be free and perform the indicated summation in (56), we obtain a hypergeometric-based formula that appears to reproduce not only the formulas in Appendix E for integer α, but also half-integer and other nonnegative fractional values (such as 1/4,2/3) of α.Dunkl argued that fork>-α and n=1,2,3,…(59)Qk+n,α=Qk,α+Qk+1,α-Qk,α+Qk+2,α-Qk+1,α+⋯+Qk+n,α-Qk+n-1,α=Qk,α+Q-α,α∑i=0n-1Hα,k+α+1+i.Taking the limit as n→∞(60)12=Qk,α+Q-α,α∑i=0∞Hα,k+α+1+i=Qk,α+Q-α,αHα,k+α+1∑i=0∞Hα,k+α+1+iHα,k+α+1;thus(61)Qk,α=12-Q-α,αHα,k+α+1∑i=0∞Hα,k+α+1+i1iHα,k+α+1i!.(Let us point the reader to an interesting partial matching between entries of the hypergeometric function and arguments of the gamma functions.) The resultant master formula takes the form (62)Qk,α=12-α20α+8k+11Γ5α+2k+2Γ3α+k+3/2Γ2α+k+3/24πΓ5α+2k+7/2Γα+k+2Γ4α+k+2×F651,52α+k+1,52α+k+32,2α+k+32,3α+k+32,52α+k+198α+k+2,4α+k+2,52α+k+74,52α+k+94,52α+k+118;1.The value 1/2 from which these terms are subtracted itself has an interesting provenance. It was obtained by conducting the sum indicated in (56), not over j from 0 to α+k as indicated there, but over j from 0 to ∞; that is, Q(-α,α)∑j=0∞H(α,j). (The Q(k,α) formula can then be recovered by subtracting the sum over j from α+k+1 to ∞; that is, Q(-α,α)∑j=α+k+1∞H(α,j).) This resulted in the expression (cf. http://math.stackexchange.com/questions/1872364/prove-that-a-certain-hypergeometric-function-assumes-either-the-value-frac1)(63)Q-α,α∑j=0∞Hα,j=π3-3α-1Γ2α+3/2F543α/2,α+1/2,3α/2+1/2,3α/2+11/8,2α+1/2;3α/2+3/8,3α/2+3/4,3α/2+5/4,3α+1;1Γα+5/6Γα+7/6.For α>0 this gives us the indicated value of 1/2. Let us note that, for both this F54 function and the F65 immediately preceding, the sums of the denominator entries minus the sums of the numerator parameters equal 1/2, while if these differences had been 1, the two functions could be designated as “1/2-balanced” [34].In the notation of this section (cf. (30)),(64)Qk+1,α-Qk,α=Q-α,αHα,α+k+1=π3-3α-1αΓ3α+3/220α+8k+11Γk+2α+3/2Γk+3α+3/2Γ2k+5α+22Γα+1/2Γα+5/6Γα+7/6Γk+α+2Γk+4α+2Γ2k+5α+7/2.


### 11.1. Implications forP(k,α) Formula

Let us note that, for the Hilbert-Schmidt (k=0) case, apparently [9], 2Q(0,α)=P(0,α), where(65)Q0,α=12-2-4α-420α+11Γ3α+32Γ5α+2F651,2α+32,5α2+1,5α2+32,5α2+198,3α+32;α+2,5α2+118,5α2+74,5α2+94,4α+2;1Γ2αΓα+2Γ5α+72-1.Thus, any presumed “master formula” for P(k,α) (Section 12) should reduce to 2Q(0,α) for k=0 (cf. eqs. (31)–(33)). We have been investigating the use of 2Q(k,α) as an initial candidate for P(k,α), then padding out the six upper and five lower entries of the F65 function with additional pairs of entries, identical for k=0, but different for k≠0. Then, for k=0, the initial candidate would be recovered. (The somewhat interesting “1/2-balanced” property, mentioned above, or some k-free counterpart of it would, then, be lost.) Initial limited numerical investigations along these lines have been somewhat disappointing, as they appeared to indicate that the best fits would be obtained for pairs of padded entries withequal coefficients of k. Also, fits to values of P(k,α) did not seem to be improved through the padding strategy.However, another considerably more interesting approach along similarly motivated lines was, then, developed. We mapped the parameterk in the Q(k,α) function to βk, so that for k=0 the original function would be recovered, no matter the specific value of β. We evaluated the transformed functions by seeing how well they fit the series of (known) eight values P(k,k), k=5,…,12. For the original β=1, the figure-of-merit for the fit was 0.7703536. This figure rather dramatically decreases/improves as β increases, reaching a near minimum of 0.0479732 for β=11/2 (and 0.108008 for β=5 and 0.153828 for β=6.) The implications of this phenomenon will be further investigated. Perhaps it might be of value tocombine the last two (padding and scaling of k) strategies.


### 11.2. Conjectured Identity

In relation to (63), Dunkl formulated the conjecture(66)F543α2,α+12,3α2+12,3α2+118,2α+12;3α2+38,3α2+34,3α2+54,3α+1;1=322274αΓα+5/6Γα+7/6Γα+3/4Γα+5/4. To avoid zero denominators, it is necessary that α>-1/8. For α=0, the value is 1, while the sum is rational for α=n, n+1/2, n=0,1,2,….In response to this conjecture, C. Koutschan wrote: “The5F4 sum fits into the class of identities that can be done with Zeilberger’s algorithm. I attach a Mathematica notebook with some computations. More precisely, using the creative telescoping method, my program finds a linear recurrence equation that is satisfied by the 5F4 sum. It is a trivial calculation to verify that also the right-hand side satisfies the same recurrence. As you remark, both sides give 1 for α=0. We can conclude that the identity holds for all α in N.” However, cases where α is neither an integer nor half-integer still require attention. (Gasper has commented that the F54 function isnot a special case of the formulas in his paper with Rahman [35].)


## 11.1. Implications forP(k,α) Formula

Let us note that, for the Hilbert-Schmidt (k=0) case, apparently [9], 2Q(0,α)=P(0,α), where(65)Q0,α=12-2-4α-420α+11Γ3α+32Γ5α+2F651,2α+32,5α2+1,5α2+32,5α2+198,3α+32;α+2,5α2+118,5α2+74,5α2+94,4α+2;1Γ2αΓα+2Γ5α+72-1.Thus, any presumed “master formula” for P(k,α) (Section 12) should reduce to 2Q(0,α) for k=0 (cf. eqs. (31)–(33)). We have been investigating the use of 2Q(k,α) as an initial candidate for P(k,α), then padding out the six upper and five lower entries of the F65 function with additional pairs of entries, identical for k=0, but different for k≠0. Then, for k=0, the initial candidate would be recovered. (The somewhat interesting “1/2-balanced” property, mentioned above, or some k-free counterpart of it would, then, be lost.) Initial limited numerical investigations along these lines have been somewhat disappointing, as they appeared to indicate that the best fits would be obtained for pairs of padded entries withequal coefficients of k. Also, fits to values of P(k,α) did not seem to be improved through the padding strategy.However, another considerably more interesting approach along similarly motivated lines was, then, developed. We mapped the parameterk in the Q(k,α) function to βk, so that for k=0 the original function would be recovered, no matter the specific value of β. We evaluated the transformed functions by seeing how well they fit the series of (known) eight values P(k,k), k=5,…,12. For the original β=1, the figure-of-merit for the fit was 0.7703536. This figure rather dramatically decreases/improves as β increases, reaching a near minimum of 0.0479732 for β=11/2 (and 0.108008 for β=5 and 0.153828 for β=6.) The implications of this phenomenon will be further investigated. Perhaps it might be of value tocombine the last two (padding and scaling of k) strategies.


## 11.2. Conjectured Identity

In relation to (63), Dunkl formulated the conjecture(66)F543α2,α+12,3α2+12,3α2+118,2α+12;3α2+38,3α2+34,3α2+54,3α+1;1=322274αΓα+5/6Γα+7/6Γα+3/4Γα+5/4. To avoid zero denominators, it is necessary that α>-1/8. For α=0, the value is 1, while the sum is rational for α=n, n+1/2, n=0,1,2,….In response to this conjecture, C. Koutschan wrote: “The5F4 sum fits into the class of identities that can be done with Zeilberger’s algorithm. I attach a Mathematica notebook with some computations. More precisely, using the creative telescoping method, my program finds a linear recurrence equation that is satisfied by the 5F4 sum. It is a trivial calculation to verify that also the right-hand side satisfies the same recurrence. As you remark, both sides give 1 for α=0. We can conclude that the identity holds for all α in N.” However, cases where α is neither an integer nor half-integer still require attention. (Gasper has commented that the F54 function isnot a special case of the formulas in his paper with Rahman [35].)


## 12. Master Formula Investigation forP(k,α)

Appendix  A in [1] considered the possibility of developing a master formula for the total separability probability P(k,α), associated with the determinantal inequality ρPT>0 (cf. (4)–(6)). It now clearly seems appropriate to reexamine those results (Appendix F) in terms of the striking hypergeometric-based formula (Section 11) we have obtained for the partial separability probability Q(k,α), associated with the determinantal inequality ρPT>ρ.In the earlier study [1], the formulas took the form of 1 minus terms involving polynomials in k and gamma functions, while above the interesting such terms have been subtracted from 1/2. So, conjecturally there exists a tightly related analogue of the results reported in Section 11 for P(k,α). (Dunkl did note the qualitative difference that “the ratio (1/2-Qk+1,α)/(1/2-Qk,a) tends to 1 as k→∞ but (1-Pk+1,α)/(1-Pk,a) tends to 16/27.”)In investigating these matters, we have found that, for our set of computedP(k,α), α=1,…,47, the number and location of the consecutive negative roots (Section 10.3) are precisely the same (55) as for Q(k,α) (Section 10.3). (There strangely appears to be a sole exception to this rule for α=3, where there are five such roots for Q(k,3) and six such for P(k,3), with P(-3,3) anomalously equalling 0.) However, in the P(k,α) situation, the component polynomials are of degree 4α-2, while in the Q(k,α) setting the corresponding polynomials are of considerably smaller degree α+α-1/2-1, so we are faced with a greater number of coefficients to determine.Here, is the equation we have solved to determine—based on [1, Appendix  A]—formulas for P(k,α) for α=1,…,47. The c’s are (nonnegative integer) coefficients we fitted to exact values obtained using the Legendre-polynomial density-approximation routine of Provost [18]. (The first 15,761 of the moments (8) were employed.)(67)Pk,α=1-28α+2k+1k-α+1/3-3Γk+3α+32Γ2k+5α+2Γk+3α+α+13+1kα+1/3+3∑i=13α-α+1/3+α+1/2-3ci+1ki-1+c1+kk3α+α+1/2πΓ2αΓ3k+10α+2Γk+2α+α+12+1-1.


### 12.1. The Ratios(P(k+1,α)-P(k,α))/(Q(k+1,α)-Q(k,α))

In AppendixG, we show a number of formulas we have generated for the differences between the formulas for P(k,α) for successive values of k, in relation to the earlier Q(k,α)-based formulas shown in Appendix C. (A stark contrast occurs, with the formulas k=-2,…,-10 initially yielding [“biproper”] rational functions—with equal-degree numerators and [zero constant term] denominators [the degrees satisfying a certain difference equation]—and, then, difference equations for k>-2.) So, it appears that the quest for a general P(k,α) formula could be successfully addressed by employing the same framework as in the Q(k,α) case, by modifying the H(α,j) function to incorporate the new terms shown in Appendix C and their extensions to k, in general. We see an evident relation between the coefficients of the y[1+α] terms in the difference equations in Appendix G and the six hypergeometric upper parameters described in Section 3.1.2 in the pattern of two 6’s and four 5’s. Also, the coefficients of the y[α] terms appear related to the six hypergeometric lower parameters described in Section 3.1.1.Further, in AppendixH we show the ratios as functions of k, rather than of α.


#### 12.1.1. Solution of Difference Equation for(P(1,α)-P(0,α))/(Q(1,α)-Q(0,α))

We have been successfully able to solve the second difference equation recorded (in two forms) in AppendixG. The initial solution consisted of a large (multipage) output with numerous hypergeometric functions (again with argument 27/64). (In Appendix I, we show the Maple counterpart, provided by Carl Love (http://math.stackexchange.com/questions/1903720/what-solution-does-maple-give-to-this-difference-equation), of our Mathematica solution. There is an implicit [unperformed] summation in it.) The solution naturally broke into the sum of two parts. For the first part—using high-precision numerics, rationalizations, and the FindSequenceFunction command—we were able to obtain the (hypergeometric-free) formula(68)53-3α-182α+15α+37/10α9/10α1α11/10α13/10α3/2α20α+112/5α3/5α4/5α5/6α7/6α6/5α.Remarkably, when this term was multiplied by the function (which comprises the denominator of the ratio), examples of which are shown in Appendix C and formulated in (64),(69)Q1,α-Q0,α=π2-4α33α+15-5α-320α+11Γα+5/6Γα+7/6Γ5α+2ΓαΓα+7/10Γα+9/10Γα+11/10Γα+13/10Γ2α+3,the product simplified to the form 4α(5α+3)/9(α+1). So, we can consider this term to be the first of two parts of a formula for P(1,α)-P(0,α). Now, in quest of the remaining term, when we formed a new difference equation for just the second part, we obtained a new solution, again naturally breaking into the sum of two parts. Now, the first part—previously given by (68)—was zero, and the new second part was given by precisely the same difference equation as original, but for the single change of the initial value (at α=1) from y[1]=158/31=474/93 to y[1]=-4102/93.


### 12.2.X-States Counterpart

In AppendixJ we show the analogue of the P(k,α) formulas for the “toy” model ofX-states [30, 31]. One feature to be immediately noted is that the arguments of the indicated hypergeometric functions are −1. Another is that, for half-integer α’s, P(k,α) yields rational values, while PX-states(k,α) yields value of the form 1 minus rational numbers divided by π2.


### 12.3. Use of Consecutive Negative Roots

We have noted that bothQ(k,α) and P(k,α) have roots at consecutive negative values of k (Section 10.3). If we examine the (limiting) values of P(k,α) for k immediately (one) below the end of the consecutive series, we find that they satisfy the relation(70)P-14-1α-1α10α+1-1,α=3α5α+2-1sin⁡πα/24α+1+cos⁡πα2.(This might serve as a “starting point” analogous to the use ((56), (57)) of Q(-α,α)). For the analogous set of Q(-(1/4)(-1)α((-1)α(10α+1)-1),α)’s, the real parts appear to be 1/2 for even α and -1/4 for odd α, with the imaginary parts given by(71)IQ-14-1α-1α10α+1-1,α=-3-1α20-1α+3α+5-1α+74π400α2+80α+3.Dunkl has observed that the sequence generated by (70) is really two interspersed sequences, one for odd and one for even values of α. They can be represented as f(2α)=(-1)α and f(2α+1)=(-1)α(15α2+18α+5)/(2α+2)=(-1)α(15α/2+1/(α+1)+3/2).


### 12.4. Settingk so That the F65 Parameters in the Q(k,α) Formula Are Zero

It appeared to be an exercise of interest to setk in P(k,α) so that, in turn, one of the five variable upper and lower parameters in the F65 function in the formula for Q(k,α) would equal zero. We now enumerate those such scenarios, for which we were able to construct formulas.Fork→-2-4α, we found that(72)Pk,α=324α-15α+2Γ2α+3/2π3α+2Γ2α+2+14.As already observed, since we have consecutive roots descending downward from-1-α, for k=-2-α, we have P(k,α)=0.Further, we found that, in the limitk→-1-5α/2, (73)Pk,α=-∞,α≡1mod⁡4,Pk,α=-1,α≡2mod⁡4,Pk,α=∞,α≡3mod⁡4,Pk,α=1,α≡0mod⁡4.Also, for k→-3/2-5α/2, for even α(74)Pk,α=-∞,α≡2mod⁡4,Pk,α=∞,α≡0mod⁡4,and for odd α(75)Pk,α=-iiα3α5α+2-14α+1.Additionally, along similar investigative lines, we have theP(k,α) formulas in Appendix L, particularly elegantly (k→-1-4α),(76)Pk,α=14316αΓ2α+1/2πΓ2α+1+1.


### 12.5. Formula forP(-α,α)

Also, eventually (after having computedP(k,α) for α=1,…,49), we were able to obtain the formula (not as explicit as that for Q(-α,α)) shown in Appendix M for P(-α,α).


### 12.6. TwoP(k,α) Formulas Involving the Lerch Transcendent

In the limitk-→-1/2-5α/2, we found forevenα,(77)Pk,α=iα3Φ-1,1,α2+110+5Φ-1,1,α2+16-3Φ-1,1,α2+310-3Φ-1,1,α2+710+5Φ-1,1,α2+56+3Φ-1,1,α2+910-2Φ-1,1,α+1215π-1.Here, the Lerch transcendent Φ(z,s,b)=∑i=0∞zi/(i+b)s. In the same limit, we have foroddα,(78)Qk,α=3π/5Γα/2+7/10Γα/2+9/10Γα/2+11/10Γα/2+13/10Γα+1/28Γα/2+3/5Γα/2+4/5Γα/2+1Γα/2+6/5Γα/2+7/5.Next, in the limitk-→-5α/2, we found foroddα,(79)Pk,α=-ieiπα/2-40α+5α+2α5Φ-1,1,α2+13+3Φ-1,1,α2+25-3Φ-1,1,α2+35-5Φ-1,1,α2+23+3Φ-1,1,α2+45+2Φ-1,1,α2+1+3Φ-1,1,α2+65+1215πα5α+2-1,While forevenα,(80)Qk,α=3π5/25-5α/2-235α+22Γ5α/2+288Γα/2+7/10Γα/2+9/10Γα/2+11/10Γα/2+13/10Γα+3/2. We note the similarities in integer coefficients between the two Lerch-based P(k,α) formulas and then by now familiar occurrences (Sections 3.1 and 12.1.1, Appendix A) of simple fractions with denominators that are multiples of five and six.


### 12.7. Rules for Leading Coefficients of the Polynomialspα(k)

In AppendixN we show for i=1,…,10, the first of the rules we have developed for the leading coefficients of the polynomials pα(k) given in the formula above (1) for P(k,α), having been normalized to monic form (the original leading degree-(4α-2) coefficient being 28α+1/2α-1!). (For convenience, we drop this k4α-2 term, leaving us with degree-(4α-3) polynomials.) We note that these resultant polynomials are of degree 2i. Now, we can make the interesting observation (essentially putting the polynomial in Horner form) that their leading (highest power) coefficients are given (in descending order) by the rules:(81)C1=17/2iΓi+1,C2=2-i-217i-21109-497i3Γi,C3=2-i-517i-4ii247009i-1370262+3942323-113087349Γi.Also, C4 is the product of(82)-2-i-717i-6i-1i405Γi+1,613817365i4-5492491130i3+30016283027i2-173872269670i+542508998592.Further, C5 is the product of(83)2-i-1117i-8i-1i1215Γi+1,305067230405i6-4403156498055i5+38051293414691i4-325978342903557i3+2137571940201488i2-8722204904328012i+13657232612174832.Continuing, C6 is the product of(84)-2-i-1317i-10i-2i-1i25515Γi+1,212265778915799i7-4033760477145378i6+46257531538470350i5-526319720165886192i4+5002806671861237555i3-35895786322816308558i2+169446873953910154824i-385892347895176978944,while C7 is the product of(85)2-i-1617i-12i-2i-1i1148175Γi+1,527480460605760515i9-14061542253335879085i8+216128338841103270330i7-3070915881213672409050i6+39074939804872696010811i5-414647891239558549971645i4+3466800379462987766973880i3-20874814527662001270399420i2+78054176824402526959936464i-118165465673929410155118720.So, an obvious important challenge would be to find the common formula generating these results. (The pattern of [negative] integer exponents of 2, that is, 0,2,5,7,11,13,16, is yielded by sequence A004134 “denominators in expansion of (1-x)-1/4 are 2a(n)” of the On-Line Encyclopedia of Integer Sequences.)Let us make the observation that the constant (lowest-order) coefficient in the polynomialpk(α) in the formula for P(k,α) in (1) is equal to (1-2Q0,α)/G(0,α).


## 12.1. The Ratios(P(k+1,α)-P(k,α))/(Q(k+1,α)-Q(k,α))

In AppendixG, we show a number of formulas we have generated for the differences between the formulas for P(k,α) for successive values of k, in relation to the earlier Q(k,α)-based formulas shown in Appendix C. (A stark contrast occurs, with the formulas k=-2,…,-10 initially yielding [“biproper”] rational functions—with equal-degree numerators and [zero constant term] denominators [the degrees satisfying a certain difference equation]—and, then, difference equations for k>-2.) So, it appears that the quest for a general P(k,α) formula could be successfully addressed by employing the same framework as in the Q(k,α) case, by modifying the H(α,j) function to incorporate the new terms shown in Appendix C and their extensions to k, in general. We see an evident relation between the coefficients of the y[1+α] terms in the difference equations in Appendix G and the six hypergeometric upper parameters described in Section 3.1.2 in the pattern of two 6’s and four 5’s. Also, the coefficients of the y[α] terms appear related to the six hypergeometric lower parameters described in Section 3.1.1.Further, in AppendixH we show the ratios as functions of k, rather than of α.


### 12.1.1. Solution of Difference Equation for(P(1,α)-P(0,α))/(Q(1,α)-Q(0,α))

We have been successfully able to solve the second difference equation recorded (in two forms) in AppendixG. The initial solution consisted of a large (multipage) output with numerous hypergeometric functions (again with argument 27/64). (In Appendix I, we show the Maple counterpart, provided by Carl Love (http://math.stackexchange.com/questions/1903720/what-solution-does-maple-give-to-this-difference-equation), of our Mathematica solution. There is an implicit [unperformed] summation in it.) The solution naturally broke into the sum of two parts. For the first part—using high-precision numerics, rationalizations, and the FindSequenceFunction command—we were able to obtain the (hypergeometric-free) formula(68)53-3α-182α+15α+37/10α9/10α1α11/10α13/10α3/2α20α+112/5α3/5α4/5α5/6α7/6α6/5α.Remarkably, when this term was multiplied by the function (which comprises the denominator of the ratio), examples of which are shown in Appendix C and formulated in (64),(69)Q1,α-Q0,α=π2-4α33α+15-5α-320α+11Γα+5/6Γα+7/6Γ5α+2ΓαΓα+7/10Γα+9/10Γα+11/10Γα+13/10Γ2α+3,the product simplified to the form 4α(5α+3)/9(α+1). So, we can consider this term to be the first of two parts of a formula for P(1,α)-P(0,α). Now, in quest of the remaining term, when we formed a new difference equation for just the second part, we obtained a new solution, again naturally breaking into the sum of two parts. Now, the first part—previously given by (68)—was zero, and the new second part was given by precisely the same difference equation as original, but for the single change of the initial value (at α=1) from y[1]=158/31=474/93 to y[1]=-4102/93.


## 12.1.1. Solution of Difference Equation for(P(1,α)-P(0,α))/(Q(1,α)-Q(0,α))

We have been successfully able to solve the second difference equation recorded (in two forms) in AppendixG. The initial solution consisted of a large (multipage) output with numerous hypergeometric functions (again with argument 27/64). (In Appendix I, we show the Maple counterpart, provided by Carl Love (http://math.stackexchange.com/questions/1903720/what-solution-does-maple-give-to-this-difference-equation), of our Mathematica solution. There is an implicit [unperformed] summation in it.) The solution naturally broke into the sum of two parts. For the first part—using high-precision numerics, rationalizations, and the FindSequenceFunction command—we were able to obtain the (hypergeometric-free) formula(68)53-3α-182α+15α+37/10α9/10α1α11/10α13/10α3/2α20α+112/5α3/5α4/5α5/6α7/6α6/5α.Remarkably, when this term was multiplied by the function (which comprises the denominator of the ratio), examples of which are shown in Appendix C and formulated in (64),(69)Q1,α-Q0,α=π2-4α33α+15-5α-320α+11Γα+5/6Γα+7/6Γ5α+2ΓαΓα+7/10Γα+9/10Γα+11/10Γα+13/10Γ2α+3,the product simplified to the form 4α(5α+3)/9(α+1). So, we can consider this term to be the first of two parts of a formula for P(1,α)-P(0,α). Now, in quest of the remaining term, when we formed a new difference equation for just the second part, we obtained a new solution, again naturally breaking into the sum of two parts. Now, the first part—previously given by (68)—was zero, and the new second part was given by precisely the same difference equation as original, but for the single change of the initial value (at α=1) from y[1]=158/31=474/93 to y[1]=-4102/93.


## 12.2.X-States Counterpart

In AppendixJ we show the analogue of the P(k,α) formulas for the “toy” model ofX-states [30, 31]. One feature to be immediately noted is that the arguments of the indicated hypergeometric functions are −1. Another is that, for half-integer α’s, P(k,α) yields rational values, while PX-states(k,α) yields value of the form 1 minus rational numbers divided by π2.


## 12.3. Use of Consecutive Negative Roots

We have noted that bothQ(k,α) and P(k,α) have roots at consecutive negative values of k (Section 10.3). If we examine the (limiting) values of P(k,α) for k immediately (one) below the end of the consecutive series, we find that they satisfy the relation(70)P-14-1α-1α10α+1-1,α=3α5α+2-1sin⁡πα/24α+1+cos⁡πα2.(This might serve as a “starting point” analogous to the use ((56), (57)) of Q(-α,α)). For the analogous set of Q(-(1/4)(-1)α((-1)α(10α+1)-1),α)’s, the real parts appear to be 1/2 for even α and -1/4 for odd α, with the imaginary parts given by(71)IQ-14-1α-1α10α+1-1,α=-3-1α20-1α+3α+5-1α+74π400α2+80α+3.Dunkl has observed that the sequence generated by (70) is really two interspersed sequences, one for odd and one for even values of α. They can be represented as f(2α)=(-1)α and f(2α+1)=(-1)α(15α2+18α+5)/(2α+2)=(-1)α(15α/2+1/(α+1)+3/2).


## 12.4. Settingk so That the F65 Parameters in the Q(k,α) Formula Are Zero

It appeared to be an exercise of interest to setk in P(k,α) so that, in turn, one of the five variable upper and lower parameters in the F65 function in the formula for Q(k,α) would equal zero. We now enumerate those such scenarios, for which we were able to construct formulas.Fork→-2-4α, we found that(72)Pk,α=324α-15α+2Γ2α+3/2π3α+2Γ2α+2+14.As already observed, since we have consecutive roots descending downward from-1-α, for k=-2-α, we have P(k,α)=0.Further, we found that, in the limitk→-1-5α/2, (73)Pk,α=-∞,α≡1mod⁡4,Pk,α=-1,α≡2mod⁡4,Pk,α=∞,α≡3mod⁡4,Pk,α=1,α≡0mod⁡4.Also, for k→-3/2-5α/2, for even α(74)Pk,α=-∞,α≡2mod⁡4,Pk,α=∞,α≡0mod⁡4,and for odd α(75)Pk,α=-iiα3α5α+2-14α+1.Additionally, along similar investigative lines, we have theP(k,α) formulas in Appendix L, particularly elegantly (k→-1-4α),(76)Pk,α=14316αΓ2α+1/2πΓ2α+1+1.


## 12.5. Formula forP(-α,α)

Also, eventually (after having computedP(k,α) for α=1,…,49), we were able to obtain the formula (not as explicit as that for Q(-α,α)) shown in Appendix M for P(-α,α).


## 12.6. TwoP(k,α) Formulas Involving the Lerch Transcendent

In the limitk-→-1/2-5α/2, we found forevenα,(77)Pk,α=iα3Φ-1,1,α2+110+5Φ-1,1,α2+16-3Φ-1,1,α2+310-3Φ-1,1,α2+710+5Φ-1,1,α2+56+3Φ-1,1,α2+910-2Φ-1,1,α+1215π-1.Here, the Lerch transcendent Φ(z,s,b)=∑i=0∞zi/(i+b)s. In the same limit, we have foroddα,(78)Qk,α=3π/5Γα/2+7/10Γα/2+9/10Γα/2+11/10Γα/2+13/10Γα+1/28Γα/2+3/5Γα/2+4/5Γα/2+1Γα/2+6/5Γα/2+7/5.Next, in the limitk-→-5α/2, we found foroddα,(79)Pk,α=-ieiπα/2-40α+5α+2α5Φ-1,1,α2+13+3Φ-1,1,α2+25-3Φ-1,1,α2+35-5Φ-1,1,α2+23+3Φ-1,1,α2+45+2Φ-1,1,α2+1+3Φ-1,1,α2+65+1215πα5α+2-1,While forevenα,(80)Qk,α=3π5/25-5α/2-235α+22Γ5α/2+288Γα/2+7/10Γα/2+9/10Γα/2+11/10Γα/2+13/10Γα+3/2. We note the similarities in integer coefficients between the two Lerch-based P(k,α) formulas and then by now familiar occurrences (Sections 3.1 and 12.1.1, Appendix A) of simple fractions with denominators that are multiples of five and six.


## 12.7. Rules for Leading Coefficients of the Polynomialspα(k)

In AppendixN we show for i=1,…,10, the first of the rules we have developed for the leading coefficients of the polynomials pα(k) given in the formula above (1) for P(k,α), having been normalized to monic form (the original leading degree-(4α-2) coefficient being 28α+1/2α-1!). (For convenience, we drop this k4α-2 term, leaving us with degree-(4α-3) polynomials.) We note that these resultant polynomials are of degree 2i. Now, we can make the interesting observation (essentially putting the polynomial in Horner form) that their leading (highest power) coefficients are given (in descending order) by the rules:(81)C1=17/2iΓi+1,C2=2-i-217i-21109-497i3Γi,C3=2-i-517i-4ii247009i-1370262+3942323-113087349Γi.Also, C4 is the product of(82)-2-i-717i-6i-1i405Γi+1,613817365i4-5492491130i3+30016283027i2-173872269670i+542508998592.Further, C5 is the product of(83)2-i-1117i-8i-1i1215Γi+1,305067230405i6-4403156498055i5+38051293414691i4-325978342903557i3+2137571940201488i2-8722204904328012i+13657232612174832.Continuing, C6 is the product of(84)-2-i-1317i-10i-2i-1i25515Γi+1,212265778915799i7-4033760477145378i6+46257531538470350i5-526319720165886192i4+5002806671861237555i3-35895786322816308558i2+169446873953910154824i-385892347895176978944,while C7 is the product of(85)2-i-1617i-12i-2i-1i1148175Γi+1,527480460605760515i9-14061542253335879085i8+216128338841103270330i7-3070915881213672409050i6+39074939804872696010811i5-414647891239558549971645i4+3466800379462987766973880i3-20874814527662001270399420i2+78054176824402526959936464i-118165465673929410155118720.So, an obvious important challenge would be to find the common formula generating these results. (The pattern of [negative] integer exponents of 2, that is, 0,2,5,7,11,13,16, is yielded by sequence A004134 “denominators in expansion of (1-x)-1/4 are 2a(n)” of the On-Line Encyclopedia of Integer Sequences.)Let us make the observation that the constant (lowest-order) coefficient in the polynomialpk(α) in the formula for P(k,α) in (1) is equal to (1-2Q0,α)/G(0,α).


## 13. Concluding Remarks

The asymptotic analyses reported here and those in studies of Aubrun et al. [4, 36, 37] both employ Hilbert-Schmidt and (more generally) random induced measures (cf. [38]). However, contrastingly, we chiefly consider asymptotics as the Dyson-index-like parameter α→∞ (cf. [39, 40]), while they implicitly are concerned with the standard case of α=1 and large numbers of qubits. Perhaps some relation exists, however, between their high-dimensional findings and the quite limited set of asymptotics we have presented above (Sections 7.2, 7.3, and 8.1.2), pertaining to the dimensional index k→∞.A strong, intriguing theme in the analyses presented above has been the repeated occurrence of the interesting constantz=27/64=(3/4)3. Let us note that Guillera in his article “A New Ramanujan-Like Series for 1/π2,” applying methods related to Zeilberger’s algorithm [33], obtained a hypergeometric identity involving a sum over n from 0 to ∞ of terms involving factors of the form (27/64)n [27, sec. 3] (cf. [32, sec. 8]).Further, in a study of products of Ginibre matrices of Penson and Życzkowski, the Fuss-Catalan distributionPs(x) is represented as a sum of s generalized hypergeometric functions Fss-1, somewhat analogous to that given in (Fig. 3 in [17]), since only F76 functions are employed. These functions Ps(x) have hypergeometric arguments (ss/(s+1)(s+1))x, where s is a nonnegative integer, and have support x∈[0,ss/(s+1)(s+1)] [41, eq. (11)]. So, for s=3, ss/(s+1)s+1=27/256. (We had inquired of Hou whether the telescoping procedure might be profitably applied in such a context. He replied “the method I used only works for Fss-1 with a concrete integer s” [cf. [41, eqs. (13)–(16)]].) As an item of further curiosity, we note that, in the MathWorld entry on hypergeometric functions, the identity F21(1/3,2/3;5/6;27/32)=8/5, the argument being 27/32, is noted. (Also, cf. (57) above.)



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*Source: 9365213-2018-05-24.xml*