Charting the $q$-Askey scheme. II. The $q$-Zhedanov scheme

DOI10.1016/J.INDAG.2022.12.003arXiv2209.07995MaRDI QIDQ6411014

Publication date: 16 September 2022

Abstract: This is the second in a series of papers which intend to explore conceptual ways of distinguishing between families in the

q

-Askey scheme and uniform ways of parametrizing the families. For a system of polynomials

p_{n} (x)

in the

q

-Askey scheme satisfying

L p_{n} = h_{n} p_{n}

with

L

a second order

q

-difference operator the

q

-Zhedanov algebra is the algebra generated by operators

L

and

X

(multiplication by

x

). It has two relations in which essentially five coefficients occur. Vanishing of one or more of the coefficients corresponds to a subfamily or limit family of the Askey-Wilson polynomials. An arrow from one family to another means that in the latter family one more coefficient vanishes. This yields the

q

-Zhedanov scheme given in this paper. The

q

-hypergeometric expression of

p_{n} (x)

can be interpreted as an expansion of

p_{n} (x)

in terms of certain Newton polynomials. In our previous paper arXiv:2108.03858 we used Verde-Star's clean parametrization of such expansions and we obtained a

q

-Verde-Star scheme, where vanishing of one or more of these parameters corresponds to a subfamily or limit family. The actions of the operators

L

and

X

on the Newton polynomials can be expressed in terms of the Verde-Star parameters, and thus the coefficients for the

q

-Zhedanov algebra can be expressed in terms of these parameters. There are interesting differences between the

q

-Verde-Star scheme and the

q

-Zhedanov scheme, which are discussed in the paper.

Mathematics Subject Classification ID

Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) (33D45) Connections of basic hypergeometric functions with quantum groups, Chevalley groups, (p)-adic groups, Hecke algebras, and related topics (33D80)

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