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Generalized Bochner theorem: Characterization of the Askey–Wilson polynomials

✍ Scribed by Luc Vinet; Alexei Zhedanov

Publisher
Elsevier Science
Year
2008
Tongue
English
Weight
184 KB
Volume
211
Category
Article
ISSN
0377-0427

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✦ Synopsis

Assume that there is a set of monic polynomials P n (z) satisfying the second-order difference equation

where z(s), A(s), B(s), C(s) are some functions of the discrete argument s and N may be either finite or infinite. The irreducibility condition A(s -1)C(s) = 0 is assumed for all admissible values of s. In the finite case we assume that there are N + 1 distinct grid points z(s), s = 0, 1, . . . , N such that z(i) = z(j ), i = j . If N =∞ we assume that the grid z(s) has infinitely many different values for different values of s. In both finite and infinite cases we assume also that the problem is non-degenerate, i.e., n = m , n = m. Then we show that necessarily: (i) the grid z(s) is at most quadratic or q-quadratic in s; (ii) corresponding polynomials P n (z) are at most the Askey-Wilson polynomials corresponding to the grid z(s). This result can be considered as generalizing of the Bochner theorem (characterizing the ordinary classical polynomials) to generic case of arbitrary difference operator on arbitrary grids.

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