Charting the 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 p_n (x)in the -Askey scheme satisfying Lp_n = h_np_n with 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 (Contemporary Math. 780) 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.