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§18.27 q-Hahn Class

Contents

§18.27(i) Introduction

The q-hypergeometric OP’s comprise the q-Hahn class OP’s and the Askey–Wilson class OP’s (§18.28). For the notation of q-hypergeometric functions see §§17.2 and 17.4(i).

The q-Hahn class OP’s comprise systems of OP’s {pn(x)}, n=0,1,,N, or n=0,1,2,, that are eigenfunctions of a second-order q-difference operator. Thus

18.27.1 A(x)pn(qx)+B(x)pn(x)+C(x)pn(q-1x)=λnpn(x),

where A(x), B(x), and C(x) are independent of n, and where the λn are the eigenvalues. In the q-Hahn class OP’s the role of the operator d/dx in the Jacobi, Laguerre, and Hermite cases is played by the q-derivative 𝒟q, as defined in (17.2.41). A (nonexhaustive) classification of such systems of OP’s was made by Hahn (1949). There are 18 families of OP’s of q-Hahn class. These families depend on further parameters, in addition to q. The generic (top level) cases are the q-Hahn polynomials and the big q-Jacobi polynomials, each of which depends on three further parameters.

All these systems of OP’s have orthogonality properties of the form

18.27.2 xXpn(x)pm(x)|x|vx=hnδn,m,

where X is given by X={aqy}yI+ or X={aqy}yI+{-bqy}yI-. Here a,b are fixed positive real numbers, and I+ and I- are sequences of successive integers, finite or unbounded in one direction, or unbounded in both directions. If I+ and I- are both nonempty, then they are both unbounded to the right. Some of the systems of OP’s that occur in the classification do not have a unique orthogonality property. Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval.

Here only a few families are mentioned. They are defined by their q-hypergeometric representations, followed by their orthogonality properties. For other formulas, including q-difference equations, recurrence relations, duality formulas, special cases, and limit relations, see Koekoek et al. (2010, Chapter 14). See also Gasper and Rahman (2004, pp. 195–199, 228–230) and Ismail (2005, Chapters 13, 18, 21).

§18.27(ii) q-Hahn Polynomials

18.27.3 Qn(x)=Qn(x;α,β,N;q)=ϕ23(q-n,αβqn+1,xαq,q-N;q,q),
n=0,1,,N.
18.27.4 y=0NQn(q-y)Qm(q-y)(αq,q-N;q)y(αβq)-y(q,β-1q-N;q)y=hnδn,m,
n,m=0,1,,N.

For hn see Koekoek et al. (2010, Eq. (14.6.2)).

§18.27(iii) Big q-Jacobi Polynomials

18.27.5 Pn(x;a,b,c;q)=ϕ23(q-n,abqn+1,xaq,cq;q,q),

and

18.27.6 Pn(α,β)(x;c,d;q)=cnq-(α+1)n(qα+1,-qα+1c-1d;q)n(q,-q;q)n×Pn(qα+1c-1dx;qα,qβ,-qαc-1d;q).

The orthogonality relations are given by (18.27.2), with

18.27.7 pn(x)=Pn(x;a,b,c;q),
18.27.8 X={aq+1}=0,1,2,{cq+1}=0,1,2,,
18.27.9 vx=(a-1x,c-1x;q)(x,bc-1x;q),
0<a<q-1, 0<b<q-1, c<0,

and

18.27.10 pn(x)=Pn(α,β)(x;c,d;q)
18.27.11 X={cq}=0,1,2,{-dq}=0,1,2,,
18.27.12 vx=(qx/c,-qx/d;q)(qα+1x/c,-qβ+1x/d;q),
α,β>-1, c,d>0.

For hn see Koekoek et al. (2010, Eq. (14.5.2)).

§18.27(iv) Little q-Jacobi Polynomials

18.27.13 pn(x)=pn(x;a,b;q)=ϕ12(q-n,abqn+1aq;q,qx).
18.27.14 y=0pn(qy)pm(qy)(bq;q)y(aq)y(q;q)y=hnδn,m,
0<a<q-1,b<q-1 .

For hn see Koekoek et al. (2010, Eq. (14.12.2)).

Bounds for the extreme zeros are given in Driver and Jordaan (2013).

§18.27(v) q-Laguerre Polynomials

18.27.15 Ln(α)(x;q)=(qα+1;q)n(q;q)nϕ11(q-nqα+1;q,-xqn+α+1).

The measure is not uniquely determined:

18.27.16 0Ln(α)(x;q)Lm(α)(x;q)xα(-x;q)dx=(qα+1;q)n(q;q)nqnh0(1)δn,m,
α>-1,

where h0(1) is given in Koekoek et al. (2010, Eq. (14.21.2)), and

18.27.17 y=-Ln(α)(cqy;q)Lm(α)(cqy;q)qy(α+1)(-cqy;q)=(qα+1;q)n(q;q)nqnh0(2)δn,m,
α>-1, c>0,

where h0(2) is given in Koekoek et al. (2010, Eq. (14.21.3)).

Bounds for the extreme zeros are given in Driver and Jordaan (2013).

§18.27(vi) Stieltjes–Wigert Polynomials

18.27.18 Sn(x;q)==0nq2(-x)(q;q)(q;q)n-=1(q;q)nϕ11(q-n0;q,-qn+1x).

(Sometimes in the literature x is replaced by q12x.)

§18.27(vii) Discrete q-Hermite I and II Polynomials

Discrete q-Hermite I

18.27.21 hn(x;q)=(q;q)n=0n/2(-1)q(-1)xn-2(q2;q2)(q;q)n-2=xnϕ02(q-n,q-n+1-;q2,x-2q2n-1).
18.27.22 =0(hn(q;q)hm(q;q)+hn(-q;q)hm(-q;q))(q+1,-q+1;q)q=(q;q)n(q,-1,-q;q)qn(n-1)/2δn,m.

Discrete q-Hermite II

18.27.23 h~n(x;q)=(q;q)n=0n/2(-1)q-2nq(2+1)xn-2(q2;q2)(q;q)n-2=xnϕ12(q-n,q-n+10;q2,-x-2q2).
18.27.24 =-(h~n(cq;q)h~m(cq;q)+h~n(-cq;q)h~m(-cq;q))q(-c2q2;q2)=2(q2,-c2q,-c-2q;q2)(q,-c2,-c-2q2;q2)(q;q)nqn2δn,m,
c>0.

(For discrete q-Hermite II polynomials the measure is not uniquely determined.)