Charting the q-Askey scheme. II. The q-Zhedanov scheme · arXivDesk
2209.07995Sep 16, 2022v3: 23 pages, 1 figure; dedicated to Jaap Korevaar on the occasion of his 100th birthday; Indag. Math., article in press; first paragraph of Introduction added; minor corrections; a few references added
Charting the q-Askey scheme. II. The q-Zhedanov scheme
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 pn(x) in the q-Askey scheme satisfying Lpn=hnpn
Nearby in the stack
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
pn(x)
can be interpreted as an expansion of
pn(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.