# Linearization Coefficients for Some Basic Hypergeometric Polynomials

Authors: Hamza Chaggara; Mohammed Mabrouk
Journal: Journal of Mathematics
 (2022)

---

## Abstract

In this paper, we give a simple and original method based on inverse relation to express explicitly the linearization coefficients for some general classes of basic hypergeometric polynomial set in terms of the basic Kampé de Fériet function. We use symbolic computation algorithms, namely,q-Multisum to find recurrence relations for the resulting linearization coefficients and qsum17 to solve some from the obtained recurrence relations. In some cases, the linearization coefficients are reduced to hypergeometric functions or hypergeometric terms.


---

## Body

## 1. Introduction

The linearization problem consists in finding the coefficientsLijk in the expression as follows:(1)PixQjx=∑k=0i+jLijkRkx,with Pnn≥0,Qnn≥0, and Rnn≥0 are three polynomial sets degPn=degQn=degRn=n. In the special case, Pn=Qn=Rn, the problem is reduced to the standard linearization or Clebsch–Gordan type problem [1].The linearization coefficients play an important role not only in many problems in pure and applied mathematics ranging from combinatorics to group theory in relation with Clebsch–Gordan coefficients [2] but also in mathematical physics and in physical and quantum chemical applications [1, 3].To solve this problem, many researchers have proposed various methods: recursively [4–7], analytically [8–10], by combinatoric approach [11] and by operational rules: dual sequences and lowering operators [12, 13]. All these methods have been intensively tested and applied so that polynomials could check some properties.In the standard case and whenPnn≥0 is a system of orthogonal polynomials, the linearization coefficient Lijk can be computed as the integral of the triple product PiPjPk, like in the of Gegenbauer polynomials. Usually, as for the Jacobi polynomials, explicit formulas expressing Lijk are not available.For a large class of polynomials, a lot of researchers have worked to find a one-index recurrence relation satisfied by the linearization coefficients. In many situations, those recurrence relations have been solved algorithmically using Mathematica and Maple software. Even if the explicit solution is not given by the computer, properties of those coefficients can be obtained from the obtained recurrence relation, namely, positivity, asymptotic limits relations, and so on [14–16].The purpose of this paper is to express the linearization coefficients for a general class of basic polynomials containing, as particular cases, some basic hypergeometric polynomials in and beyond theq-Askey scheme. Basic (or q-) series and basic (or q-) polynomials especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences (see, for details [17], pp. 351–352).Our main technical is to use explicit formulas for structural identities of basic polynomials systems and a general inversion formula derived from Carlitz’s inverse relation (Lemma3.2), and then by composition, we get the desired coefficients. Moreover, we should mention here that the use of inversion in order to solve expansions problems was firstly considered by Rainville [18] for classical orthogonal polynomials and then adopted by many other authors [8, 19, 20].As an application, we derive the linearization coefficients for some known polynomials basis and we give one-index recurrence relation satisfied by the obtained coefficients, using for this purpose specific computer algebra algorithms theq-Multisum package of Mathematica program, implemented by A. Riese [21]. In some cases, recurrence relations can be solved by using the q-hypergeometric summation package qsum17, a q-analogue of Zeilberger’s algorithm, implemented in Maple by W. Koepf and T. Sprenger [22]. The closed analytical formulas for the obtained linearization coefficients will be expressed by means of terminating double basic hypergeometric series, basic hypergeometric functions which, in some cases, can be evaluated as a basic hypergeometric terms.The paper is organized as follows:In Section2, we recall some definitions and notations as well as some useful identities.In Section3, we prove our main results (Theorems 3.1–3.3) and we give some useful consequences and special cases.Section4 is devoted to application of the obtained results to some known families of polynomials, namely, the q-Pochhammer symbols, the Little q-Laguerre, and so on. In the standard case, we give recurrence relations satisfied by the linearization coefficients, which can be solved by symbolic computations.


## 2. Basic Definitions and Notations

In this work, we adopt the commonly used conventions and notations for the basic (orq-) series and the basic (or q-) polynomials. For the convenience of the reader, we first provide a summary of the mathematical notations, basic properties, and identities to be used in this paper. We refer to the general references (see, for example, [23–26]) for the definitions and notations.The Pochhammer symbol is defined by(2)an=∏k=0n−1a+k,ifn=1,2,3,…,1,ifn=0.Theq-Pochhammer symbol, called also the q-shifted factorial, is defined as(3)a;qn=∏k=0n−11−aqk,ifn=1,2,3,…,1,ifn=0.It is clear that(4)limq⟶1−qa;qk1−qk=ak.Theq-binomial coefficient and the binomial coefficient are, respectively, given by(5)nkq=q;qnq;qkq;qn−k,nk=n!k!n−k!.Some useful identities are(6)q−n;qk=−1kq;qnq;qn−kqkk−12−nk,n≥k,(7)a;qn−k=a;qnqkk+1/2q1−n/a;qk−akqnk,n≥k,(8)aqn;qka;qk=aqn;qna;qn.The basic hypergeometric series orq-hypergeometric series are defined by [25]:(9)rϕsarbs;q;x=∑k=0∞ar;qkbs;qk−1kqkk−1/21+s−rxkq;qk,where(10)ar;qk=a1;qka2;qk⋯ar;qk,k=0,1,2,…,provided that no zero factors appear in the denominator on the right-hand side, i.e., none of the denominator parametersbkk=1s has the form q−m with m∈ℕ∗.λ:=λnn≥0,μ:=μnn≥0 and ν:=νnn≥0 designate three sequences of nonzero complex numbers.Letxnn and ynn be two sequences of complex numbers. An inverse relation is a pair of equations of the following form [27]:(11)yn=∑k=0nλnan,kμkxk,xn=∑k=0nμna^n,kλkyk.A general class of polynomials is defined by(12)Pnx;a,λ|q=∑k=0nq−n;qkaqn;qkλkqkxk,which represents aq-analogue of(13)Pnx,a,λ=∑k=0n−nka+nkλkxk.The basisx⊖aqnn≥0 is defined by(14)x−aqn=a−ax−aq…a−aqn−1=−anqnn−1/2ϕ10q−n−;q;qxa.The basic Kampé de Fériet function is a two-variable generalization of the basic hypergeometric series. It is given by(15)ϕt:u;vp:s;rap:bs;cr;q:x,yαt:βu;γv;i,j,k=∑m=0∞∑n=0∞ap;qm+nbs;qmcr;qnαt;qm+nβu;qmγv;qnxmynqim/2+jn/2+kmnq;qmq;qn.For this double basic hypergeometric series, see [28] and the references therein.This basic function is aq-analogue of the Kampé de Fériet function Ft:u;vp:s;r introduced by Srivastava and Karlsson [17]. Indeed,(16)limq⟶1−ϕt:u;vp:s;rqap:qbs;qcr;q:x1−qt+u−p−s+1,y1−qt−p+v−r+1qαt:qβu;qγv;i,j,k=Ft:u;vp:s;rap:bs;crαt:βu;γv,x,y,where(17)Ft:u;vp:s;rap:bs;crαt:βu;γv,x,y=∑m=0∞∑n=0∞apn+mbsncrmαtn+mβunγvmxmynm!n!,where apn=∏i=1pain.


## 3. Linearization Coefficients

In this section, our interest is to find the explicit formula of the linearization coefficients relating three generalized basic polynomials defined by (12).Theorem 3.1.
The linearization coefficients in(18)Pix;a,λ|qPjx;α,μ|q=∑k=0i+jLijkPkx;c,ν|q,are given by(19)Lijk=∑r=0i∑s=0j−1kqkk−1/2q−i;qi−raqi;qi−rq−j;qj−sαqj;qj−scqk;qkq;qkcq2k+1;qi+j−r−s−kq;qi+j−r−s−kλi−rμj−sνi+j−r−sqi+j−r−s.To prove Theorem3.1, we need the following Carlitz’s inverse relation.Lemma 3.2.
(Theorem 4 in [27] and Formula (1.2) in [29]). Let ai and bi be two sequences of complex numbers, let q an arbitrary complex number such that ai+q−kbi≠0,i=1,2,…,k=0,1,2,…, and put(20)ψk,n,q=∏i=1nai+q−kbi.
Then, we have the following inverse relation:(21)yn=∑k=0n−1kqkk+1/2−nknkqψ−k,n,qxk,xn=∑k=0n−1kqkk−1/2nkqak+1+qkbk+1ψ−n,k+1,qyk.Proof.
of Theorem3.1. Let an,k=q−n;qnaqn;qkqk. In view of relations (6)–(8), we obtain(22)an,k=−1ka;qnqkk+1/2−nknkqaqk,qnq;qka;qk.
According to (11) and (21) with λn=1/a;qn and μn=1/q;qna;qn, we get(23)a^n,k=−1kqkk−1/2nkq1−aq2kψ−n,k+1,qa;qkq;qna;qn=nkq−1kqkk−1/21−aq2kaqn;qk+1a;qkq;qna;qn=−1kqkk−1/21−aq2ka;qkq;qkq;qn−ka;qnaqn;qk+1=−1kqkk−1/21−aq2kq;qkq;qn−kaqk;qn+1=−1kqkk−1/21−aq2kq;qkq;qn−kaqk;qkaq2k;qn−k+1=−1kq1/2kk−1aqk;qkq;qkaq2k+1;qn−kq;qn−k.
Then, the inversion formula for the basic polynomials (12) follows(24)xl=∑k=0l−1kqkk−1/2cqk;qkq;qkcq2k+1;ql−kq;ql−kPkx;c,ν|qνl.
To find the linearization coefficients in (18), we make the product of two polynomials (12):(25)Pix;a,λ|qPjx;α,μ|q=∑r=0iq−i;qraqi;qrλrqrxr∑s=0jq−j;qsaqj;qsαsqsxs=∑r=0iq−i;qi−raqi;qi−rλi−rqi−r∑s=0jq−j;qj−saqj;qj−sαj−sqj−sxi+j−r−s,then we use (24), which yields to the following linearization formula:(26)Pix;a,λ|qPjx;α,μ|q=∑k=0i+j∑r=0i∑s=0j−1kqkk−1/2q−i;qi−raqi;qi−rcqk;qkq;qkcq2k+1;qi+j−r−s−kq−j;qj−sαqj;qj−sq;qi+j−r−s−kλi−rμj−sνi+j−r−sqi+j−r−sPkx;c,ν|q.Formula (18) includes many linearization formulas of polynomials sets: q-shifted, Little q-Jacobi, Little q-Laguerre, Stieltjes-Wigert, and some d-orthogonal basic hypergeometric polynomials: d-orthogonal little q-Laguerre type, d-orthogonal big q-Laguerre type, and d-orthogonal q-Laguerre type [19].Theorem 3.3.
The linearization coefficients in the formula(27)ϕ21q−i,aqib;q;qx2ϕ1q−j,αqjβ;q;qx=∑k=0i+jLijk2ϕ1q−k,cqkd;q;qx,are given by(28)Liji+j−k=−1kqkk+1/2i+jkqqij−ki+jaqi;qiαqj;qjd;qi+jb;qiβ;qjcqi+j−k;qi+j−kcq2i+j−k+1;qk×ϕ2:1;12:2;2q−k,qk−2i−2jc:q−i,q1−ib;q−j,q1−jβ;q:bcadq,βcαdqq−i−j,q1−i−jd:q1−2ia;q1−2jα;0,0,0Proof.
By choosingλr=1/b;qrq;qr,νs=1/β;qsq;qs,μl=1/d;qlq;ql, in formula (14) and with the relations (6)–(8), we get(29)Liji+j−k=−1i+j−kqi+j−ki+j−k−1/2∑r=0i∑s=0jq−i,aqi;qi−rq−j,αqj;qj−scqi+j−k,q;qi+j−kcq2i+j−k+1,q;qk−r−s×d;qi+j−r−sb;qi−rβ;qj−sq;qi+j−r−sq;qi−rq;qj−sqi+j−r−s=−1kqi+j−ki+j−k−1/2q−i;qrqri+1−ii+1/2q;qrq−j;qsqs1+j−jj+1/2q;qsaqi;qiqrr+1/2q1−2i/a;qr−aqirqri×αqj;qjqss+1/2q1−2j/αs−αqjsqjsqk−2i−2j/c;qr+s−cq2i+j−k+1r+sqkr+scqi+j−k;qi+j−kcq2i+j−k+1;qkqr+sr+s+1/2q1−i/b;qr−brqrib;qiqrr+1/2×q−k;qr+sq;qi+jqr+sk−i−jq−i−j;qr+sq;qi+j−kq;qkq1−j/β;qs−βsqsjβ;qjqss+1/2d;qi+jqr+sr+s+1/2qi+j−r−sq1−i−j/d;qr+s−dr+sqr+si+j=−1kqkk+1/2+ij−ki+ji+jkqaqi;qiαqj;qjd;qi+jb;qiβ;qjcqi+j−k;qi+j−kcq2i+j−k+1;qk×∑r=0i∑s=0jq−k;qr+sqk−2i−2j/c;qr+sq−i;qrq1−i/b;qrq−j;qsq1−j/β;qsq;qrq;qsq−i−j;qr+sq1−i−j/d;qr+sq1−2i/arq1−2i/a;qs×bcqadrcβqdαs.
Then, (28) follows.Some particular cases are worthy to note.Case 1.
If we substitutea=α=c⟶0 in (21), we obtain the linearization formula:(30)ϕ11q−ib;q;qx1ϕ1q−jβ;q;qx=∑k=0i+jLijk1ϕ1q−kd;q;qx,with(31)Liji+j−k=−1kqkk+1/2−ki+ji+jkq×ϕ2:0;01:2;2q−k:q−i,q1−ib;q−j,q1−jβ;q:qk−jbd,qk−iβdq−i−j,q1−i−jd:−;−;0,0,0.Case 2.
The linearization coefficients in(32)Pix;a,λPjx;α,μ=∑k=0i+jLijkPkx,c,νare given by(33)Lijk=∑r=0i∑s=0j−1k−ii−ra+ii−r−jj−sα+jj−sk!c+kki+j−r−s−k!c+2k+1i+j−r−s−kλi−rμj−sνi+j−r−s,where Pix;a,λ is given by (13).
This result can be derived from (18) by replacing a,b,α,β,c,andd, respectively, by qa,qb,qα,qβ, qc, andqd, by letting q⟶1− and using (4) and (16).Case 3.
The linearization coefficients in(34)F21−i,a+ib;x2F1−j,α+jβ;x=∑k=0i+jLijkF21−k,c+kd;x,are given by:(35)Liji+j−k=−1ki+jkc+2i+j−ka+iiα+jjc+i+j−ki+jdi+jbiβj×F2:2;22:1;1−k,k−2i−2j−c:−i,1−i−b;−j,1−j−β;−i−j:1−2i−a;1−2j−α;1,1.
The linearization formula (34) is a special case of formula (3.5) stated in [13] via the generating functions and the Lagrange expansion formula.
Note that, many well-known polynomial sets and their various generalizations throughout the literature are included in family (13): Shifted Jacobi polynomials, Srivastava-Pathan polynomials, generalized Bessel polynomials and generalized Rice polynomials [13], Laguerre polynomials, and d-Laguerre polynomials [30], thereby we can get the corresponding linearization coefficients for all the aforementioned families.


## 4. Applications

In this section, we apply (28) and (31) to solve some linearization problems for classical basic polynomials. We use computer algebra systems (Mathematica and Maple) to find recurrence relations satisfied by the obtained coefficients and we try to reduce the corresponding basic Kampé de Fériet function into a single sum or a hypergeometric term. Some of the obtained formulas were already obtained in many other works [12, 20, 31] using, for this purpose, q-difference equations, structure relations, orthogonality, generating functions, and so forth.


### 4.1. Linearization Coefficients of the Basisx⊖aqnn≥0

Proposition 4.1.
The basisx⊖aqnn≥0 defined by (10) fulfills the following linearization relation [20, 31]:(36)x⊖aqix⊖aqj=∑k=0mini,j−akqkk−1/2q;qiq;qjq;qkq;qj−kq;qi−kx⊖aqi+j−k.Proof.
By applying Theorem3.3 and the basic hypergeometric representation (10), we get the following linearization formula:(37)x⊖aqix⊖bqj=∑k=0i+jLijkx⊖cqk,with(38)Liji+j−k=cki+jkqϕ1:0;01:1;1q−k:q−i;q−j;q;aqk−jc,bqk−icq−i−j:−;−;0;0,0.
If we putSk:=Liji+j−k and by use of the q-Multisum package,(39)qk−cqi+j−cqi+1+j+cq1+k+aqi+1+k+bqj+1+k+cq2+i+j+k−aq2+2k−bq2+2kS1+k−qqi+j−qkcqj−aqkcqi−bqkSk+q1+2k−1+q2+kS2+k=0.
The latter is reduced, for the special casea=b=c, to the following simpler recurrence relation:(40)a2qqi−qkqj−qkqi+j−qkSk−qk+2−1q2k+1Sk+2−aqkqi+j+k+2−qi+j−qi+j+1+qi+k+1+qj+k+1+qk+1−2q2k+2Sk+1=0,with the initial values, S−1:=0andS0=1.
Via the qrecsolve command of the qsum17 package, we find the exact solution of the recurrence relation (40):(41)Sk=q−j;qkq−i;qk−aqi+jkq−k2q;qk,and therefore the following linearization formula holds:(42)x⊖aqix⊖aqj=∑k=0mini,j−aqi+jkq−k2q−j;qkq−i;qkq;qkx⊖aqi+j−k.
Then, (36) follows from identity (7).Proposition 4.2.
For the Al-Salam-Carlitz I polynomials set [23] defined by(43)Unax;q=−anqn2ϕ21q−n,x−10;q;qxa,the following linearization formula holds [20]:(44)x⊖1qix⊖1qj=∑k=0i+jai+j−ki+jkq3ϕ1​​​​​​qk−i−j,q−i,q−jq−i−j​​​​;q;qi+j−kaUkax;q.Proof.
Combining the following linearization formula, which can be derived from (28) by replacing k by i+j−k,(45)x⊖1qix⊖1qj=∑k=0i+j−qi+ji+j−kq−i+j−k2q−j;qi+j−kq−i;qi+j−kq;qi+j−kx⊖1qk,with the connection formula given in [19]:(46)x⊖1qk=∑m=0kak−mkmqUmax;q,and identity (8) yields(47)x−1qix−1qj=∑k=0i+j−qi+ji+j−kq−i+j−k2q−j;qi+j−kq−i;qi+j−kq;qi+j−k∑m=0kak−mkmqUmax;q=∑m=0i+j∑k=0i+j−m−qi+ji+j−k−mq−i+j−k−m2q−j;qi+j−k−mq−i;qi+j−k−mq;qi+j−k−makk+mmqUmax;q=∑m=0i+j∑r=0i+j−m−qi+jrq−r2q−j;qrq−i;qrq;qi+j−rq;qrq;qi+j−k−mq;qrq;qmai+j−m−rUmax;q=∑m=0i+jai+j−mi+jm​q∑r=0i+j−m−1rq−r2qm−i−j;qrq−j;qrq−i;qrq;qrq−i−j;qrqi+j−marUmax;q.
Then, by virtue of (9), we derive (44).


### 4.2. Linearization Coefficients of theq-Pochhammer Basis

Proposition 4.3.
Theq-Pochhammer basis x;qnn≥0 fulfills the following linearization formula [20, 31]:(48)x;qix;qj=∑k=0mini,j−1kqkk−12−ijq;qiq;qjq;qkq;qi−kq;qj−kx;qi+j−k.Proof.
Theq-Pochhammer basis have the expression:(49)x;qn=1ϕ0​​q−n−​​​​;q;qnx.
Then, by applying Theorem3.3, we find the linearization formula(50)x;qix;qj=∑k=0i+jLijkx;qk,with(51)Liji+j−k=−1kqkk−1/2−iji+jkqϕ1:0;01:1;1q−k:q−i;q−j;q;q,qq−i−j:−;−;0;0,0.
Using again theq-Multisum algorithm, for Sk:=Liji+j−kqij, we obtain the following two-order recurrence relation:(52)qk−qi+j+k+1+2qi+j−qi+k+1−qj+k+1+q2k+1+q2k+2−qkSk+1+Skqi−qkqj−qkqi+j−qk−qk+2−1q2k+1Sk+2=0,with S−1:=0andS0=1.
Then, by using the qsum17 package, we find the solution of the recurrence relation:(53)Sk=q−i;qkq−j;qk−qi+jkq−k2q;qk.
Finally, according to (7), we deduce (48).Remark.
In view of the limit relation (2), we can recover the linearization formula for the Pochhammer polynomials xnn≥0 [12]:(54)xixj=∑k=0mini,j−1k−ik−jkk!xi+j−k.Proposition 4.4.
For the bigq-Jacobi polynomials Pnx;a,b,c;q defined by [24]:(55)Pnx;a,b,c;q=3ϕ2q−n,abqn+1,xaq,cq;q;q,we have the following linearization formula:(56)x;qix;qj=∑k=0i+j−1kqkk−1/2i+jkqaq;qi+jcq;qi+jabq2k+2;qi+j−kabqk+1;qk×4ϕ3​​​​​​q−i,q−j,qk−i−j,q−i−j−k−1abq−i−j,q−i−ja,q−i−jc​​​​;q;bqcPkx;a,b,c;q.Proof.
Combining the linearization formula,(57)x;qix;qj=∑k=0i+j−qi+ji+j−kq−i+j−k2q−j;qi+j−kq−i;qi+j−kq;qi+j−kx;qk,with the inversion formula of the big q-Jacobi polynomials [19]:(58)x;qk=∑m=0k−1mkmqqmm−1/2aq,cq;qkabq2m+2;qk−mabqm+1;qmPmx;a,b,c;q,and according to (6) and (7), we obtain the following:(59)x;qix;qj=∑k=0i+j−qi+ji+j−kq−i+j−k2q−j;qi+j−kq−i;qi+j−kq;qi+j−k∑m=0k−1mkmqqm2×aq,cq;qkabq2m+2;qk−mabqm+1;qmPmx;a,b,c;q=∑m=0i+j∑k=0i+j−m−qi+ji+j−k−mq−i+j−k−m2q−j;qi+j−k−mq−i;qi+j−k−mq;qi+j−k−m−1mqm2k+mmq×aq;qk+mcq;qk+mabq2m+2;qkabqm+1;qmPmx;a,b,c;q=∑m=0i+j∑r=0i+j−m−qi+jrq−r2q−j;qrq−i;qrq;qr−1mqm2i+j−rmq×aq;qi+j−rcq;qi+j−rabq2m+2;qi+j−r−mabqm+1;qmPmx;a,b,c;q=∑m=0i+j−1mqm2i+jmqaq;qi+jcq;qi+jabqm+1;qmabq2m+2;qi+j−m∑r=0i+j−mbqcr×qm−i−j;qrq−i;qrq−j;qrq−i−j−m−1/ab;qrq−i−j;qrq−i−j/a;qrq−i−j/c;qrq;qrPmx;a,b,c;q.
Hence, by (9), we obtain formula (56).


### 4.3. Linearization ofq-Laguerre Polynomials

Theq-Laguerre polynomials defined by [24]:(60)Lnαx,q=qα+1;qnq;qn1ϕ1q−nq;q;−qn+α+1x,have the following linearization formula:(61)Liαx,qLjβx,q=∑k=0mini,jLiji+j−kLi+j−kγx,q,with(62)Liji+j−k=−1kqkk−12i+jjqqγ+i+j−k+1;qkq;qkqiα−γ−j+jβ−γ−i×ϕ2:0;01:2;2q−k:q−i,q−i−α;q−j,q−j−β;q;q,qq−i−j,q−i−j−γ:−;−;0;0,0.For the standard case,α=β=γ in (61), where we replace x by q−αx and let α⟶∞, we obtain the linearization formula for the Stieltjes–Wigert polynomials as follows:(63)Six;qSjx;q=∑k=0mini,j−1kqkk−1/2q−2ijq;qki+jjq×ϕ1:0;01:1;1q−k:q−i;q−j;q;q1+j,q1+iq−i−j:−;−;0;0,−1Skx,q,where the Stieltjes–Wigert polynomials are defined by [24]:(64)Snx;q=1q;qn1ϕ1q−n0;q;−qn+1x.By using theq-Multisum package, we obtain a seven-order recurrence equation. Unfortunately, the qsum package did not yield any q-hypergeometric term solution. Therefore, we have an option to look for mixed recurrence equations.(65)−qi+j+1Li+1,j+1k+1+qkLi,j+1k+qkLi+1,jk−qkLi,jk+qi+j+2−qkLi+1,j+1k=0,with the initial conditions:(66)Li,0k=0,k=0,…i−1,L0,jk=0,k=0…j−1,Li,ji+j+m=0,m=1,2,…andLi,ji+j=1.Similar mixed recurrence relations can be also obtained forq-Laguerre polynomials, little q-Laguerre polynomials, Al-Salam-Carlitz I polynomials, and so on.Figure 1
Stress concentration in weld location.


## 4.1. Linearization Coefficients of the Basisx⊖aqnn≥0

Proposition 4.1.
The basisx⊖aqnn≥0 defined by (10) fulfills the following linearization relation [20, 31]:(36)x⊖aqix⊖aqj=∑k=0mini,j−akqkk−1/2q;qiq;qjq;qkq;qj−kq;qi−kx⊖aqi+j−k.Proof.
By applying Theorem3.3 and the basic hypergeometric representation (10), we get the following linearization formula:(37)x⊖aqix⊖bqj=∑k=0i+jLijkx⊖cqk,with(38)Liji+j−k=cki+jkqϕ1:0;01:1;1q−k:q−i;q−j;q;aqk−jc,bqk−icq−i−j:−;−;0;0,0.
If we putSk:=Liji+j−k and by use of the q-Multisum package,(39)qk−cqi+j−cqi+1+j+cq1+k+aqi+1+k+bqj+1+k+cq2+i+j+k−aq2+2k−bq2+2kS1+k−qqi+j−qkcqj−aqkcqi−bqkSk+q1+2k−1+q2+kS2+k=0.
The latter is reduced, for the special casea=b=c, to the following simpler recurrence relation:(40)a2qqi−qkqj−qkqi+j−qkSk−qk+2−1q2k+1Sk+2−aqkqi+j+k+2−qi+j−qi+j+1+qi+k+1+qj+k+1+qk+1−2q2k+2Sk+1=0,with the initial values, S−1:=0andS0=1.
Via the qrecsolve command of the qsum17 package, we find the exact solution of the recurrence relation (40):(41)Sk=q−j;qkq−i;qk−aqi+jkq−k2q;qk,and therefore the following linearization formula holds:(42)x⊖aqix⊖aqj=∑k=0mini,j−aqi+jkq−k2q−j;qkq−i;qkq;qkx⊖aqi+j−k.
Then, (36) follows from identity (7).Proposition 4.2.
For the Al-Salam-Carlitz I polynomials set [23] defined by(43)Unax;q=−anqn2ϕ21q−n,x−10;q;qxa,the following linearization formula holds [20]:(44)x⊖1qix⊖1qj=∑k=0i+jai+j−ki+jkq3ϕ1​​​​​​qk−i−j,q−i,q−jq−i−j​​​​;q;qi+j−kaUkax;q.Proof.
Combining the following linearization formula, which can be derived from (28) by replacing k by i+j−k,(45)x⊖1qix⊖1qj=∑k=0i+j−qi+ji+j−kq−i+j−k2q−j;qi+j−kq−i;qi+j−kq;qi+j−kx⊖1qk,with the connection formula given in [19]:(46)x⊖1qk=∑m=0kak−mkmqUmax;q,and identity (8) yields(47)x−1qix−1qj=∑k=0i+j−qi+ji+j−kq−i+j−k2q−j;qi+j−kq−i;qi+j−kq;qi+j−k∑m=0kak−mkmqUmax;q=∑m=0i+j∑k=0i+j−m−qi+ji+j−k−mq−i+j−k−m2q−j;qi+j−k−mq−i;qi+j−k−mq;qi+j−k−makk+mmqUmax;q=∑m=0i+j∑r=0i+j−m−qi+jrq−r2q−j;qrq−i;qrq;qi+j−rq;qrq;qi+j−k−mq;qrq;qmai+j−m−rUmax;q=∑m=0i+jai+j−mi+jm​q∑r=0i+j−m−1rq−r2qm−i−j;qrq−j;qrq−i;qrq;qrq−i−j;qrqi+j−marUmax;q.
Then, by virtue of (9), we derive (44).


## 4.2. Linearization Coefficients of theq-Pochhammer Basis

Proposition 4.3.
Theq-Pochhammer basis x;qnn≥0 fulfills the following linearization formula [20, 31]:(48)x;qix;qj=∑k=0mini,j−1kqkk−12−ijq;qiq;qjq;qkq;qi−kq;qj−kx;qi+j−k.Proof.
Theq-Pochhammer basis have the expression:(49)x;qn=1ϕ0​​q−n−​​​​;q;qnx.
Then, by applying Theorem3.3, we find the linearization formula(50)x;qix;qj=∑k=0i+jLijkx;qk,with(51)Liji+j−k=−1kqkk−1/2−iji+jkqϕ1:0;01:1;1q−k:q−i;q−j;q;q,qq−i−j:−;−;0;0,0.
Using again theq-Multisum algorithm, for Sk:=Liji+j−kqij, we obtain the following two-order recurrence relation:(52)qk−qi+j+k+1+2qi+j−qi+k+1−qj+k+1+q2k+1+q2k+2−qkSk+1+Skqi−qkqj−qkqi+j−qk−qk+2−1q2k+1Sk+2=0,with S−1:=0andS0=1.
Then, by using the qsum17 package, we find the solution of the recurrence relation:(53)Sk=q−i;qkq−j;qk−qi+jkq−k2q;qk.
Finally, according to (7), we deduce (48).Remark.
In view of the limit relation (2), we can recover the linearization formula for the Pochhammer polynomials xnn≥0 [12]:(54)xixj=∑k=0mini,j−1k−ik−jkk!xi+j−k.Proposition 4.4.
For the bigq-Jacobi polynomials Pnx;a,b,c;q defined by [24]:(55)Pnx;a,b,c;q=3ϕ2q−n,abqn+1,xaq,cq;q;q,we have the following linearization formula:(56)x;qix;qj=∑k=0i+j−1kqkk−1/2i+jkqaq;qi+jcq;qi+jabq2k+2;qi+j−kabqk+1;qk×4ϕ3​​​​​​q−i,q−j,qk−i−j,q−i−j−k−1abq−i−j,q−i−ja,q−i−jc​​​​;q;bqcPkx;a,b,c;q.Proof.
Combining the linearization formula,(57)x;qix;qj=∑k=0i+j−qi+ji+j−kq−i+j−k2q−j;qi+j−kq−i;qi+j−kq;qi+j−kx;qk,with the inversion formula of the big q-Jacobi polynomials [19]:(58)x;qk=∑m=0k−1mkmqqmm−1/2aq,cq;qkabq2m+2;qk−mabqm+1;qmPmx;a,b,c;q,and according to (6) and (7), we obtain the following:(59)x;qix;qj=∑k=0i+j−qi+ji+j−kq−i+j−k2q−j;qi+j−kq−i;qi+j−kq;qi+j−k∑m=0k−1mkmqqm2×aq,cq;qkabq2m+2;qk−mabqm+1;qmPmx;a,b,c;q=∑m=0i+j∑k=0i+j−m−qi+ji+j−k−mq−i+j−k−m2q−j;qi+j−k−mq−i;qi+j−k−mq;qi+j−k−m−1mqm2k+mmq×aq;qk+mcq;qk+mabq2m+2;qkabqm+1;qmPmx;a,b,c;q=∑m=0i+j∑r=0i+j−m−qi+jrq−r2q−j;qrq−i;qrq;qr−1mqm2i+j−rmq×aq;qi+j−rcq;qi+j−rabq2m+2;qi+j−r−mabqm+1;qmPmx;a,b,c;q=∑m=0i+j−1mqm2i+jmqaq;qi+jcq;qi+jabqm+1;qmabq2m+2;qi+j−m∑r=0i+j−mbqcr×qm−i−j;qrq−i;qrq−j;qrq−i−j−m−1/ab;qrq−i−j;qrq−i−j/a;qrq−i−j/c;qrq;qrPmx;a,b,c;q.
Hence, by (9), we obtain formula (56).


## 4.3. Linearization ofq-Laguerre Polynomials

Theq-Laguerre polynomials defined by [24]:(60)Lnαx,q=qα+1;qnq;qn1ϕ1q−nq;q;−qn+α+1x,have the following linearization formula:(61)Liαx,qLjβx,q=∑k=0mini,jLiji+j−kLi+j−kγx,q,with(62)Liji+j−k=−1kqkk−12i+jjqqγ+i+j−k+1;qkq;qkqiα−γ−j+jβ−γ−i×ϕ2:0;01:2;2q−k:q−i,q−i−α;q−j,q−j−β;q;q,qq−i−j,q−i−j−γ:−;−;0;0,0.For the standard case,α=β=γ in (61), where we replace x by q−αx and let α⟶∞, we obtain the linearization formula for the Stieltjes–Wigert polynomials as follows:(63)Six;qSjx;q=∑k=0mini,j−1kqkk−1/2q−2ijq;qki+jjq×ϕ1:0;01:1;1q−k:q−i;q−j;q;q1+j,q1+iq−i−j:−;−;0;0,−1Skx,q,where the Stieltjes–Wigert polynomials are defined by [24]:(64)Snx;q=1q;qn1ϕ1q−n0;q;−qn+1x.By using theq-Multisum package, we obtain a seven-order recurrence equation. Unfortunately, the qsum package did not yield any q-hypergeometric term solution. Therefore, we have an option to look for mixed recurrence equations.(65)−qi+j+1Li+1,j+1k+1+qkLi,j+1k+qkLi+1,jk−qkLi,jk+qi+j+2−qkLi+1,j+1k=0,with the initial conditions:(66)Li,0k=0,k=0,…i−1,L0,jk=0,k=0…j−1,Li,ji+j+m=0,m=1,2,…andLi,ji+j=1.Similar mixed recurrence relations can be also obtained forq-Laguerre polynomials, little q-Laguerre polynomials, Al-Salam-Carlitz I polynomials, and so on.Figure 1
Stress concentration in weld location.


## 5. Concluding Remarks

In this work, the linearization problem associated to a general class of a basic PS has been considered. We have described a method based on a known inverse formula to obtain an explicit expression of the associated LC often in terms a terminating basic Kampé de Fériet function.The obtained analytic representation of the LC can be used to derive an associate recurrence relation satisfied by these coefficients using, for this purpose, the symbolic computing systems Maple and Mathematica.To the best of our knowledge, most of the linearization problems have been considered here for the first time. Some known formulas were recovered as particular cases. The main linearization relations (LRs) can be summarized in the following scheme.



---

*Source: 2910077-2022-04-30.xml*