This is the second in a series of papers which intend to explore conceptual ways of distinguishing between families in the -Askey scheme and uniform ways of parametrizing the families. For a system of polynomials in the -Askey scheme satisfying with a second order -difference operator the -Zhedanov algebra is the algebra generated by operators and (multiplication by ). 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 -Zhedanov scheme given in this paper.
The -hypergeometric expression of can be interpreted as an expansion of 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 -Verde-Star scheme, where vanishing of one or more of these parameters corresponds to a subfamily or limit family. The actions of the operators and on the Newton polynomials can be expressed in terms of the Verde-Star parameters, and thus the coefficients for the -Zhedanov algebra can be expressed in terms of these parameters. There are interesting differences between the -Verde-Star scheme and the -Zhedanov scheme, which are discussed in the paper.
