ANOTHER PROOF OF BAILEY’S 6 ψ6 SUMMATIONFRÉDÉRIC JOUHET∗ AND MICHAEL SCHLOSSER∗∗
Abstract. Adapting a method used by Cauchy, Bailey, Slater, and more
recently, the second author, we give a new proof of Bailey’s celebrated 6 ψ6summation formula.

1. Introduction
In [15], one of the authors presented a new proof of Ramanujan’s 1 ψ1 summation
formula (cf. [10, Appendix (II.29)]),(q, b/a, az, q/az)∞a(1.1); q, z =1 ψ1b(b, q/a, z, b/az)∞(the notation is defined at the end of this introduction), valid for |q| < 1 and|b/a| < |z| < 1. This proof used a standard method for obtaining a bilateral identity
from a unilateral terminating identity, a method already utilized by Cauchy [8] inhis second proof of Jacobi’s [13] famous triple product identity (see (2.3)), a special
case of Ramanujan’s formula (1.1).
The same method (which is referred to as “Cauchy’s method” in the sequel) had
also been exploited by Bailey [5, Secs. 3 and 6], [6], and Slater [17, Sec. 6.2].
It was conjectured in [15, Remark 3.2] that any bilateral sum can be obtained
from an appropriately chosen terminating identity by Cauchy’s method (without
appealing to analytic continuation). However, at the same place it was also pointed
out that it was already not known whether Bailey’s [5, Eq. (4.7)] very-well-poised
6 ψ6 summation (cf. [10, Appendix (II.33)]),


√
√
qa2
q√ a, −q a, b, c, d, e
; q,
6 ψ6 √a, − a, aq/b, aq/c, aq/d, aq/e bcde(q, aq, q/a, aq/bc, aq/bd, aq/be, aq/cd, aq/ce, aq/de)∞(1.2)=(q/b, q/c, q/d, q/e, aq/b, aq/c, aq/d, aq/e, a2q/bcde)∞(again, see the end of this introduction for the notation), where |q| < 1 and|a2 q/bcde| < 1, would follow from such an identity. It is maybe interesting tomention that Bailey’s 6 ψ6 summation (1.2), although it contains more parametersthan Ramanujan’s 1 ψ1 summation (1.1), does not include the latter as a special
case.
Date: December 11, 2003.
2000 Mathematics Subject Classification. 33D15.
Key words and phrases. bilateral basic hypergeometric series, q-series, Bailey’s 6 ψ6 summation,
Ramanujan’s 1 ψ1 summation.
∗ Fully supported by EC’s IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics in Europe”, while visiting the University of Vienna from February to August 2003.
∗∗ Fully supported by an APART fellowship of the Austrian Academy of Sciences.
1

2FRÉDÉRIC JOUHET AND MICHAEL SCHLOSSER

While the conjecture of [15, Remark 3.2] remains open, this paper features a
new derivation of Bailey’s 6 ψ6 summation formula using a variant of Cauchy’s
method. After explaining some notation in the end of this introduction, the proof
from [15, Sec. 3] of Ramanujan’s 1 ψ1 summation is being reviewed in Section 2,
for illustration. The starting point for the derivation of Bailey’s 6 ψ6 summation in
Section 3 is Bailey’s terminating very-well-poised 10 φ9 transformation (3.2). Afterappropriately applying Cauchy’s method to both sides of this identity, a specifictransformation of 6 ψ6 series is obtained. The resulting transformation is then iterated, after which a suitable specialization yields the desired 6 ψ6 summation.
Other proofs of Bailey’s very-well-poised 6 ψ6 summation have been given by
Bailey [5], Slater and Lakin [18], Andrews [1], Askey and Ismail [3], Askey [2], andthe second author [14].
The authors plan to give an account of Cauchy’s method applied to terminating
quadratic, cubic and quartic identities [10, Sec. 3.8] elsewhere.
Notation: It is appropriate to recall some standard notation for q-series andbasic hypergeometric series.
Let q be a fixed complex parameter (the “base”) with 0 < |q| < 1. The q-shiftedfactorial is defined for any complex parameter a byY
(1 − aq j )(a)∞ ≡ (a; q)∞ :=
j≥0

and

(a; q)∞,(aq k ; q)∞where k is any integer. Since the same base q is used throughout this paper, it may
be readily omitted (in notation) which will not lead to any confusion. For brevity,write(a1 , · · · , am )k := (a1 )k · · · (am )k ,where k is an integer or infinity. Further, recall the definition of basic hypergeometricseries,∞X(a1 , . . . , as )ka1 , . . . , as
; q, z :=zk,s φs−1b1 , . . . , bs−1
(q, b1 , . . . , bs−1 )k(a)k ≡ (a; q)k :=k=0

and of bilateral basic hypergeometric series,∞X(a1 , . . . , as )k k
a1 , . . . , as
; q, z :=z .
s ψsb 1 , . . . , bs
(b1 , . . . , bs )kk=−∞

See Gasper and Rahman’s text [10] for a comprehensive study on the theory ofbasic hypergeometric series. In particular, the computations in this paper rely onsome elementary identities for q-shifted factorials, listed in [10, Appendix I].
2. Cauchy’s method and Ramanujan’s 1 ψ1 summation
For illustration of “Cauchy’s method”, it is convenient to review the second
author’s proof [15, Sec. 3] of Ramanujan’s 1 ψ1 summation. A closely related analysis
is being applied in Section 3 to prove Bailey’s very-well-poised 6 ψ6 summation (1.2).
In Jackson’s [11] q-Pfaff–Saalschütz summation (cf. [10, Appendix (II.12)]),(c/a, c/b)na, b, q −n
; q, q =
,
(2.1)
3 φ2
c, abq 1−n /c(c, c/ab)nANOTHER PROOF OF THE 6 ψ6 SUM

3first replace n by 2n and then shift the index of summation by n such that the newsum runs from −n to n:2n(c/a, c/b)2n X (a, b, q −2n )k=qk(c, c/ab)2n(q, c, abq 1−2n /c)k
k=0

n

X
(a, b, q −2n )n(aq n , bq n , q −n )k
=
qn
qk .
1−2n
1+n
(q, c, abq/c)n(q, cq n , abq 1−n /c)kk=−nNext, replace a by aq
n
X

k=−n

−n

, and c by cq−n, respectively, to get(c/a, cq −n /b)2n (q, cq −n , abq 1−2n /c)n −n(a, bq n , q −n )kkq=q(q 1+n , c, abq 1−n /c)k(cq −n , c/ab)2n(aq −n , b, q −2n )n=(c/a)2n (q, q, c/b, bq/c)n.
(q)2n (c, q/a, b, c/ab)nNow, one may let n → ∞, assuming |c/ab| < 1 and |b| < 1, while appealing toTannery’s theorem [7] for being allowed to interchange limit and summation. This
gives
∞
X
(q, c/a, c/b, bq/c)∞(a)k  c k,=(c)k ab(c, q/a, b, c/ab)∞k=−∞where |c/a| < |c/ab| < 1. Finally, replacing b by c/az and then c by b gives (1.1).
Remark 2.1. Cauchy [8] had applied the above method to a special case of (2.1).
(After all, (2.1) was not available to him yet.) His starting point was the terminating
q-binomial theorem,
 −n

q
n
(2.2)
; q, zq = (z)n ,1 φ0−

an identity already known to Euler [9], which can be derived from (2.1) by first
doing the substitution b 7→ c/z, then c 7→ b, and then letting a → ∞ and b → 0.
As a result of “bilateralizing” (2.2) by the above procedure, Cauchy recovered
Jacobi’s [13] well-known triple product identity,∞
X(−1)k q ( 2) z k = (q, z, q/z)∞ .
k

(2.3)

k=−∞

Note that (2.3) can be obtained from (1.1) by first replacing z by z/a, and thenletting a → ∞ and b → 0.
3. Proof of Bailey’s 6 ψ6 summation formula
In order to prove Bailey’s 6 ψ6 summation formula (1.2) by Cauchy’s method, one
should start with a known terminating identity which contains enough parameters.
If one considers Jackson’s [12] summation formula [10, Appendix (II.22)],√√a, q√ a, −q a, b, c, d, a2 q n+1 /bcd, q −n√
; q, q8 φ7a, − a, aq/b, aq/c, aq/d, bcdq −n/a, aq n+1=(aq, aq/bc, aq/bd, aq/cd)n, (3.1)(aq/b, aq/c, aq/d, aq/bcd)n4FRÉDÉRIC JOUHET AND MICHAEL SCHLOSSER

it becomes apparent that at least one parameter is missing here for the given purpose. As a next step, one might consider Watson’s transformation formula of aterminating 8 φ7 into a multiple of a 4 φ3 [10, Appendix (III.18)], which involves anadditional parameter, and apply Cauchy’s method (to both sides of the transformation). In fact, this was undertaken by Bailey [6] who obtained by this procedure, andsome symmetry argument, a transformation for 2 ψ2 series (see also [10, Ex. 5.11]).
The next level object in the hierarchy of identities for (very-well-poised) basic
hypergeometric series is Bailey’s [4] transformation formula [10, Appendix (III.28)],√√a, q a, −q a, b, c, d, e, f, λaq n+1 /ef, q −n√√
; q, q10 φ9a, − a, aq/b, aq/c, aq/d, aq/e, aq/f, ef q −n/λ, aq n+1(aq, aq/ef, λq/e, λq/f )n=(aq/e, aq/f, λq/ef, λq)n√√
λ, −q λ, λb/a, λc/a, λd/a, e, f, λaq n+1 /ef, q −nλ, q √√; q, q , (3.2)
× 10 φ9λ, − λ, aq/b, aq/c, aq/d, λq/e, λq/f, ef q −n/a, λq n+1

where λ = qa2 /bcd. A standard proof of (3.2) involves two applications of (3.1),
together with an interchange of summations, see [10, Sec. 2.9]. Jackson’s summation(3.1) itself can be proved in various ways, see e. g. Slater [17, Sec. 3.3.1], or Gasperand Rahman [10, Sec. 2.6].
Note that for b, c or d → ∞, (3.2) reduces to Watson’s transformation formula.
By a further specialization one obtains Jackson’s 8 φ7 summation formula which
may also be derived directly from (3.2) by letting b = aq/c (thus λ = a/d).
To prove (1.2), start from (3.2). First, replace n by 2n, shift the index of summation by n on both sides, and obtain(a, b, c, d, e, f, λaq 2n+1 /ef, q −2n )n(1 − aq 2n )qn
(1 − a) (q, aq/b, aq/c, aq/d, aq/e, aq/f, ef q −2n/λ, aq 2n+1 )nnX(1 − aq 2n+2k )(aq n , bq n , cq n , dq n )k×
2n1+n(1 − aq ) (q, aq 1+n /b, aq 1+n /c, aq 1+n /d)kk=−n(eq n , f q n , λaq 3n+1 /ef, q −n )kqk(aq 1+n /e, aq 1+n /f, ef q −n/λ, aq 3n+1 )k(aq, aq/ef, λq/e, λq/f )2n=(aq/e, aq/f, λq/ef, λq)2n2n(λ, λb/a, λc/a, λd/a, e, f, λaq 2n+1 /ef, q −2n )n(1 − λq )qn
×(1 − λ) (q, aq/b, aq/c, aq/d, λq/e, λq/f, ef q −2n/a, λq 2n+1 )nnX(1 − λq 2n+2k ) (λq n , λbq n /a, λcq n /a, λdq n /a)k×
(1 − λq 2n ) (q 1+n , aq 1+n /b, aq 1+n /c, aq 1+n /d)k
×

k=−n

×

(eq n , f q n , λaq 3n+1 /ef, q −n )k
qk .
1+n
(λq
/e, λq 1+n /f, ef q −n /a, λq 3n+1 )kNext, replace a, c, d, e and f by aq −2n , cq −n , dq −n , eq −n and f q −n , respectively.
This givesnX(1 − aq 2k )k=−n(aq −n , bq n , c, d, e, f, λaq n+1 /ef, q −n )kqk(1 − a) (q 1+n , aq 1−n /b, aq/c, aq/d, aq/e, aq/f, ef q −n/λ, aq n+1 )kANOTHER PROOF OF THE 6 ψ6 SUM

5(1 − aq −2n ) (aq 1−n /e, aq 1−n /f, ef q −2n /λ, aq)n(1 − a)(aq −2n , b, cq −n , dq −n )n(aq 1−2n , aq/ef, λq 1−n /e, λq 1−n /f )2n
×(aq 1−n /e, aq 1−n /f, λq/ef, λq 1−2n )2n(1 − λ) (λq −2n , λb/a, λcq −n /a, λdq −n /a)n×
(1 − λq −2n ) (λq 1−n /e, λq 1−n /f, ef q −2n /a, λq)n
=

×

n
X
(1 − λq 2k )k=−n(λq −n , λbq n /a, λc/a, λd/a, e, f, λaq n+1 /ef, q −n )kqk(1 − λ) (q 1+n , aq 1−n /b, aq/c, aq/d, λq/e, λq/f, ef q −n/a, λq n+1 )k=×(λq/e, λq/f, aq, λb/a, aq/λc, aq/λd, q/a, aq/ef )n(aq/e, aq/f, b, λq, q/c, q/d, q/λ, λq/ef )nnX(1 − λq 2k )k=−n(1 − λ)(λq −n , λbq n /a, λc/a, λd/a, e, f, λaq n+1 /ef, q −n )k
qk .
1+n
(q
, aq 1−n /b, aq/c, aq/d, λq/e, λq/f, ef q −n/a, λq n+1 )kNow, one may let n → ∞, assuming |qa2 /cdef | < 1 while appealing to Tannery’s
theorem, which yields the following transformation formula:


√
√
qa2
q√ a, −q a, c, d, e, f
√
; q,
6 ψ6
a, − a, aq/c, aq/d, aq/e, aq/f
cdef
(aq, q/a, aq/ef, aq/cd, λq/e, λq/f, aq/λc, aq/λd)∞=(aq/e, aq/f, q/c, q/d, λq, q/λ, λq/ef, b)∞
√

 √
qa2
−q λ, λc/a, λd/a, e, f
q λ,
√
√
; q,
× 6 ψ6, (3.3)λ, − λ, aq/c, aq/d, λq/e, λq/f
cdef

where λ = qa2 /bcd. In this identity, the right-hand side involves one more parameter (namely b) than the left-hand side. Note that when b = aq/cd (whence λ = a),this identity is trivial. It should also be pointed out that (3.3) is not Slater’s [16]transformation [10, Eq. (5.5.3)], the latter involving more symmetric parameters.
A possibility, of course, would be to directly specialize the extra parameter b in(3.3) such that the 6 ψ6 on the right-hand side reduces to a 6 φ5 series, which can besummed by the n → ∞ case of (3.1). However, this was not the idea of the above
derivaton of the 6 ψ6 transformation in (3.3). Indeed, several proofs of Bailey’s 6 ψ6summation which make use of the nonterminating 6 φ5 summation already exist,
see e. g. Slater and Lakin [18], Andrews [1], Askey and Ismail [3], and the second
author [14].
The clue is to iterate formula (3.3), more precisely, to apply the same transformation to the 6 ψ6 on the right-hand side of (3.3) again, with the parameters a, c,d and e, respectively, being replaced by λ, λc/a, e and λd/a. By this iteration an
additional free parameter1, say b0 , is obtained on the right-hand side. The result isthe following.


√
√
qa2
q√ a, −q a, c, d, e, f
; q,
6 ψ6 √a, − a, aq/c, aq/d, aq/e, aq/f
cdef
(aq, q/a, aq/ef, aq/cd, λq/e, aq/λd)∞=(aq/e, aq/f, q/c, q/d, λq/ef, b)∞
1On the contrary, iteration of [10, Eq. (5.5.3)] would only yield an additional redundant parameter and essentially reduce to the same identity.

6

FRÉDÉRIC JOUHET AND MICHAEL SCHLOSSER

×(aq/df, aq/ec, aλ0 q/λd, λ0 q/f, aq/λ0 c, λq/λ0 e)∞(aq/d, q/e, λ0 q, q/λ0 , aλ0 q/λdf, b0 )∞
√

 √
qa2
q λ0 , −q λ0 , λ0 c/a, λ0 e/λ, λd/a, f,;q,
× 6 ψ6 √ 0 √ 0
λ , − λ , aq/c, λq/e, λ0 aq/λd, λ0 q/f
cdef

where λ = qa2 /bcd and λ0 = aqλ/b0 ce.
Now there are two extra parameters appearing on the right-hand side. Take λ = e(thus b = qa2 /cde), and λ0 = a/c (thus b0 = q), such that the last 6 ψ6 , getting
terminated from above and from below, equals 1, which immediately establishes
Bailey’s formula (1.2) (where b has been replaced by f ).
Remark 3.1. In the above analysis, after applying Cauchy’s method to both sides of
Bailey’s 10 φ9 transformation (3.2), the resulting 6 ψ6 transformation (3.3) first wasiterated and then specialized. A natural question is what happens if one would startwith the iterate of (3.2), listed in [10, Ex. 2.19], and then apply Cauchy’s method.
In fact, the whole analysis would be similar to the one above. In particular, one
would also obtain a transformation of 6 ψ6 series (similar to but different from (3.3)
though) with an extra free parameter on the right-hand side. Again, one can iterate
the transformation to obtain a second additional parameter, and then specialize thetwo extra parameters such that the 6 ψ6 on the right-hand side reduces to a sum of
one single term only.
References[1] G. E. Andrews, Applications of basic hypergeometric functions, SIAM Rev. 16 (1974), 441–
484.
[2] R. Askey, The very well poised 6 ψ6 . II, Proc. Amer. Math. Soc. 90 (1984), 575–579.
[3] R. Askey and M. E. H. Ismail, The very well poised 6 ψ6 , Proc. Amer. Math. Soc. 77 (1979),
218–222.
[4] W. N. Bailey, An identity involving Heine’s basic hypergeometric series, J. London Math.
Soc. 4 (1929), 254–257.
[5] W. N. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. J.
Math. (Oxford) 7 (1936), 105–115.
[6] W. N. Bailey, On the basic bilateral hypergeometric series 2 ψ2 , Quart. J. Math. (Oxford) (2)1 (1950), 194–198.
[7] T. J. l’A. Bromwich, An introduction to the theory of infinite series, 2nd ed., Macmillan,
London, 1949.
[8] A.-L. Cauchy, Mémoire sur les fonctions dont plusieurs variables..., C. R. Acad. Sci. Paris
17 (1843), 523; reprinted in Oeuvres de Cauchy, Ser. 1 8, Gauthier-Villars, Paris (1893),
42–50.
[9] L. Euler, Introductio in Analysis Infinitorum, M-M Bousquet, Lausanne, 1748.
[10] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics And
Its Applications 35, Cambridge University Press, Cambridge, 1990.
[11] F. H. Jackson, Transformations of q-series, Messenger of Math. 39 (1910), 145–153.
[12] F. H. Jackson, Summation of q-hypergeometric series, Messenger of Math. 57 (1921), 101–112.
[13] C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomonti. Sumptibus fratrum Bornträger, 1829; reprinted in Jacobi’s Gesammelte Werke, vol. 1, (Reimer,
Berlin, 1881–1891), pp. 49–239; reprinted by Chelsea (New York, 1969); now distributed by
the Amer. Math. Soc., Providence, RI.
[14] M. Schlosser, A simple proof of Bailey’s very-well-poised 6 ψ6 summation, Proc. Amer. Math.
Soc. 130 (2001), 1113–1123.
[15] M. Schlosser, Abel–Rothe type generalizations of Jacobi’s triple product identity, in “Theory
and Applications of Special Functions. A Volume Dedicated to Mizan Rahman” (M. E. H.
Ismail and E. Koelink, eds.), Dev. Math., to appear.

ANOTHER PROOF OF THE 6 ψ6 SUM

7[16] L. J. Slater, General transformations of bilateral series, Quart. J. Math. (Oxford) (2) 3,
73–80.
[17] L. J. Slater, Generalized hypergeometric functions, Cambridge University Press, London/New
York, 1966.
[18] L. J. Slater and A. Lakin, Two proofs of the 6 ψ6 summation theorem, Proc. Edinburgh Math.
Soc. (2) 9 (1953–57), 116–121.
Institut Girard Desargues, Université Claude Bernard (Lyon 1), 69622 Villeurbanne
Cedex, France
E-mail address: jouhet@euler.univ-lyon1.fr
URL: http://igd.univ-lyon1.fr/home/jouhet
Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria
E-mail address: schlosse@ap.univie.ac.at
URL: http://www.mat.univie.ac.at/~schlosse