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Β§18.27 q-Hahn Class

Contents
  1. Β§18.27(i) Introduction
  2. Β§18.27(ii) q-Hahn Polynomials
  3. Β§18.27(iii) Big q-Jacobi Polynomials
  4. Β§18.27(iv) Little q-Jacobi Polynomials
  5. Β§18.27(v) q-Laguerre Polynomials
  6. Β§18.27(vi) Stieltjes–Wigert Polynomials
  7. Β§18.27(vii) Discrete q-Hermite I and II Polynomials

Β§18.27(i) Introduction

The q-hypergeometric OP’s comprise the q-Hahn class (or q-linear lattice class) OP’s and the Askey–Wilson class (or q-quadratic lattice class) OP’s (Β§18.28). Together they form the q-Askey scheme. This scheme gives a graphical representation of all families of OP’s belonging to it together with the limit relations between them, see Koekoek et al. (2010, p.Β 414).

For the notation of q-hypergeometric functions see §§17.2 and 17.4(i). Unless said otherwise, we will assume that 0<q<1. For (17.4.1) with bs=qβˆ’N, a0=qβˆ’m, and m=0,1,…,N we will use the convention that the summation on the right-hand side ends at n=m.

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⁑(q⁒x)+B⁑(x)⁒pn⁑(x)+C⁑(x)⁒pn⁑(qβˆ’1⁒x)=Ξ»n⁒pn⁑(x),

where A⁑(x), B⁑(x), and C⁑(x) are independent of n, and where the Ξ»n 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 π’Ÿq, 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 βˆ‘x∈Xpn⁑(x)⁒pm⁑(x)⁒|x|⁒vx=hn⁒δn,m,

where X is given by X={a⁒qy}y∈I+ or X={a⁒qy}y∈I+βˆͺ{βˆ’b⁒qy}y∈Iβˆ’. Here a,b are fixed positive real numbers, and I+ and Iβˆ’ are sequences of successive integers, finite or unbounded in one direction, or unbounded in both directions. If I+ and Iβˆ’ are both nonempty, then they are both unbounded to the right. In case of unbounded sequences (18.27.2) can be rewritten as a q-integral, see Β§17.2(v), and more generally Gasper and Rahman (2004, (1.11.2)). 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, ChapterΒ 14). See also Gasper and Rahman (2004, pp.Β 195–199, 228–230) and Ismail (2009, ChaptersΒ 13, 18, 21).

Β§18.27(ii) q-Hahn Polynomials

18.27.3 Qn⁑(x)=Qn⁑(x;Ξ±,Ξ²,N;q)=Ο•23⁑(qβˆ’n,α⁒β⁒qn+1,xα⁒q,qβˆ’N;q,q),
n=0,1,…,N.
18.27.4 βˆ‘y=0NQn⁑(qβˆ’y)⁒Qm⁑(qβˆ’y)⁒[Ny]q⁒(α⁒q;q)y⁒(β⁒q;q)Nβˆ’y(α⁒q)y=hn⁒δn,m,
n,m=0,1,…,N,

with

18.27.4_1 hn=(α⁒q)n⁒N1βˆ’Ξ±β’Ξ²β’q2⁒n+1⁒(α⁒β⁒qn+1;q)N+1⁒(β⁒q;q)n[Nn]q⁒(α⁒q;q)n.
18.27.4_2 limqβ†’1Qn⁑(qβˆ’x;qΞ±,qΞ²,N;q)=Qn⁑(x;Ξ±,Ξ²,N).

Β§18.27(iii) Big q-Jacobi Polynomials

18.27.5 Pn⁑(x;a,b,c;q)=Ο•23⁑(qβˆ’n,a⁒b⁒qn+1,xa⁒q,c⁒q;q,q).

Alternative definitions and notations are

18.27.6 Pn(Ξ±,Ξ²)⁑(x;c,d;q)=cn⁒qβˆ’(Ξ±+1)⁒n⁒(qΞ±+1,βˆ’qΞ±+1⁒cβˆ’1⁒d;q)n(q,βˆ’q;q)nΓ—Pn⁑(qΞ±+1⁒cβˆ’1⁒x;qΞ±,qΞ²,βˆ’qα⁒cβˆ’1⁒d;q),

and

18.27.6_5 Pn⁑(x;a,b,c,d;q)=Pn⁑(q⁒a⁒cβˆ’1⁒x;a,b,βˆ’a⁒cβˆ’1⁒d;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={a⁒qβ„“+1}β„“=0,1,2,…βˆͺ{c⁒qβ„“+1}β„“=0,1,2,…,
18.27.9 vx=(aβˆ’1⁒x,cβˆ’1⁒x;q)∞(x,b⁒cβˆ’1⁒x;q)∞,
0<a<qβˆ’1, 0<b<qβˆ’1, c<0,
18.27.9_5 hn=(βˆ’c)n⁒an+11βˆ’a⁒b⁒q2⁒n+1⁒(q;q)n⁒q(n+22)(a⁒q,c⁒q;q)n⁒(q,cβˆ’1⁒a⁒q,aβˆ’1⁒c,a⁒b⁒qn+1;q)∞(a⁒q,c⁒q,b⁒qn+1,cβˆ’1⁒a⁒b⁒qn+1;q)∞,

and

18.27.10 pn⁑(x)=Pn(α,β)⁑(x;c,d;q)
18.27.11 X={c⁒qβ„“}β„“=0,1,2,…βˆͺ{βˆ’d⁒qβ„“}β„“=0,1,2,…,
18.27.12 vx=(q⁒x/c,βˆ’q⁒x/d;q)∞(qΞ±+1⁒x/c,βˆ’qΞ²+1⁒x/d;q)∞,
Ξ±,Ξ²>βˆ’1, c,d>0.

From Big q-Jacobi to Jacobi

18.27.12_5 limqβ†’1Pn(Ξ±,Ξ²)⁑(x;c,d;q)=(c+d2)n⁒Pn(Ξ±,Ξ²)⁑(2⁒xβˆ’c+dc+d).

Β§18.27(iv) Little q-Jacobi Polynomials

18.27.13 pn⁑(x)=pn⁑(x;a,b;q)=Ο•12⁑(qβˆ’n,a⁒b⁒qn+1a⁒q;q,q⁒x)=(βˆ’b)βˆ’n⁒qβˆ’n⁒(n+1)/2⁒(q⁒b;q)n(q⁒a;q)n⁒ϕ23⁑(qβˆ’n,a⁒b⁒qn+1,q⁒b⁒xq⁒b,0;q,q).
18.27.14 βˆ‘y=0∞pn⁑(qy)⁒pm⁑(qy)⁒(b⁒q;q)y⁒(a⁒q)y(q;q)y=hn⁒δn,m,
0<a<qβˆ’1,b<qβˆ’1,

with

18.27.14_1 hn=(a⁒q)n1βˆ’a⁒b⁒q2⁒n+1⁒(q,b⁒q;q)n(a⁒q;q)n⁒(a⁒b⁒qn+1;q)∞(a⁒q;q)∞.

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

From Big q-Jacobi to Little q-Jacobi

18.27.14_2 limcβ†’βˆ’βˆžPn⁑(c⁒q⁒x;a,b,c;q)=pn⁑(x;a,b;q).
18.27.14_3 limc↑0Pn⁑(b⁒q⁒x;b,a,c;q)=(βˆ’b)n⁒qn⁒(n+1)/2⁒(q⁒a;q)n(q⁒b;q)n⁒pn⁑(x;a,b;q).

From Little q-Jacobi to Jacobi

18.27.14_4 limqβ†’1pn⁑(x;qΞ±,qΞ²;q)=n!(Ξ±+1)n⁒Pn(Ξ±,Ξ²)⁑(1βˆ’2⁒x).

Little q-Laguerre polynomials

Little q-Jacobi polynomials pn⁑(x;a,b;q) for b=0 are called little q-Laguerre or Wall polynomials:

18.27.14_5 pn⁑(x;a,0;q)=Ο•12⁑(qβˆ’n,0a⁒q;q,q⁒x).

From Little q-Laguerre to Laguerre

18.27.14_6 limqβ†’1pn⁑((1βˆ’q)⁒x;qΞ±,0;q)=n!(Ξ±+1)n⁒Ln(Ξ±)⁑(x).

Β§18.27(v) q-Laguerre Polynomials

18.27.15 Ln(Ξ±)⁑(x;q)=(qΞ±+1;q)n(q;q)n⁒ϕ11⁑(qβˆ’nqΞ±+1;q,βˆ’x⁒qn+Ξ±+1).

The measure is not uniquely determined:

18.27.16 ∫0∞Ln(Ξ±)⁑(x;q)⁒Lm(Ξ±)⁑(x;q)⁒xΞ±(βˆ’x;q)∞⁒dx=(qΞ±+1;q)n(q;q)n⁒qn⁒h0(1)⁒δn,m,
Ξ±>βˆ’1,

and

18.27.17 βˆ‘y=βˆ’βˆžβˆžLn(Ξ±)⁑(c⁒qy;q)⁒Lm(Ξ±)⁑(c⁒qy;q)⁒qy⁒(Ξ±+1)(βˆ’c⁒qy;q)∞=(qΞ±+1;q)n(q;q)n⁒qn⁒h0(2)⁒δn,m,
Ξ±>βˆ’1, c>0,

with

18.27.17_1 h0(1)=(qβˆ’Ξ±;q)∞(q;q)βˆžβ’Ξ“β‘(Ξ±+1)⁒Γ⁑(βˆ’Ξ±),
18.27.17_2 h0(2)=(q,βˆ’c⁒qΞ±+1,βˆ’cβˆ’1⁒qβˆ’Ξ±;q)∞(qΞ±+1,βˆ’c,βˆ’cβˆ’1⁒q;q)∞.

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

From q-Laguerre to Laguerre

18.27.17_3 limqβ†’1Ln(Ξ±)⁑((1βˆ’q)⁒x;q)=Ln(Ξ±)⁑(x).

Β§18.27(vi) Stieltjes–Wigert Polynomials

18.27.18 Sn⁑(x;q)=βˆ‘β„“=0nqβ„“2⁒(βˆ’x)β„“(q;q)ℓ⁒(q;q)nβˆ’β„“=1(q;q)n⁒ϕ11⁑(qβˆ’n0;q,βˆ’qn+1⁒x).

(Sometimes in the literature x is replaced by q12⁒x.)

The measure is not uniquely determined:

18.27.19 ∫0∞Sn⁑(x;q)⁒Sm⁑(x;q)(βˆ’x,βˆ’q⁒xβˆ’1;q)∞⁒dx=ln⁑(qβˆ’1)qn⁒(q;q)∞(q;q)n⁒δn,m,

and

18.27.20 ∫0∞Sn⁑(q12⁒x;q)⁒Sm⁑(q12⁒x;q)⁒exp⁑(βˆ’(ln⁑x)22⁒ln⁑(qβˆ’1))⁒dx=2⁒π⁒qβˆ’1⁒ln⁑(qβˆ’1)qn⁒(q;q)n⁒δn,m.

From Stieltjes–Wigert to Hermite

18.27.20_5 limqβ†’1(q;q)n⁒Sn⁑(qβˆ’1⁒x⁒2⁒(1βˆ’q)+1;q)(1βˆ’q2)n/2=(βˆ’1)n⁒Hn⁑(x).

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

Discrete q-Hermite I

18.27.21 hn⁑(x;q)=(q;q)nβ’βˆ‘β„“=0⌊n/2βŒ‹(βˆ’1)ℓ⁒qℓ⁒(β„“βˆ’1)⁒xnβˆ’2⁒ℓ(q2;q2)ℓ⁒(q;q)nβˆ’2⁒ℓ=xn⁒ϕ02⁑(qβˆ’n,qβˆ’n+1βˆ’;q2,xβˆ’2⁒q2⁒nβˆ’1).
18.27.22 βˆ‘β„“=0∞(hn⁑(qβ„“;q)⁒hm⁑(qβ„“;q)+hn⁑(βˆ’qβ„“;q)⁒hm⁑(βˆ’qβ„“;q))⁒(qβ„“+1,βˆ’qβ„“+1;q)∞⁒qβ„“=(q;q)n⁒(q,βˆ’1,βˆ’q;q)∞⁒qn⁒(nβˆ’1)/2⁒δn,m.

Discrete q-Hermite II

18.27.23 h~n⁑(x;q)=(q;q)nβ’βˆ‘β„“=0⌊n/2βŒ‹(βˆ’1)ℓ⁒qβˆ’2⁒n⁒ℓ⁒qℓ⁒(2⁒ℓ+1)⁒xnβˆ’2⁒ℓ(q2;q2)ℓ⁒(q;q)nβˆ’2⁒ℓ=xn⁒ϕ12⁑(qβˆ’n,qβˆ’n+10;q2,βˆ’xβˆ’2⁒q2).
18.27.24 βˆ‘β„“=βˆ’βˆžβˆž(h~n⁑(c⁒qβ„“;q)⁒h~m⁑(c⁒qβ„“;q)+h~n⁑(βˆ’c⁒qβ„“;q)⁒h~m⁑(βˆ’c⁒qβ„“;q))⁒qβ„“(βˆ’c2⁒q2⁒ℓ;q2)∞=2⁒(q2,βˆ’c2⁒q,βˆ’cβˆ’2⁒q;q2)∞(q,βˆ’c2,βˆ’cβˆ’2⁒q2;q2)∞⁒(q;q)nqn2⁒δn,m,
c>0.

For discrete q-HermiteΒ II polynomials the measure is not uniquely determined.

Limit Relations

18.27.25 limqβ†’1hn⁑((1βˆ’q2)12⁒x;q)(1βˆ’q2)n/2=2βˆ’n⁒Hn⁑(x).
18.27.26 limqβ†’1h~n⁑((1βˆ’q2)12⁒x;q)(1βˆ’q2)n/2=2βˆ’n⁒Hn⁑(x).