Polynomial Sequences Associated with the Moments of Hypergeometric Weights

Abstract

Abstract. We present some families of polynomials related to the moments of weight functions of hypergeometric type. We also consider different types of generating functions, and give several examples.

Laguerre-Freud equations for Generalized Hahn polynomials of type I

“…As we observed in [6], the finite family of polynomials that are usually called "Hahn polynomials" in the literature [33, 18.19] correspond to the special case a 1 = α + 1, a 2 = −N, b = −N − 1 − β.…”

“…They have the hypergeometric representation [42] P n (x) = (a 1 ) n (a 2 ) n (n + a 1 + a 2 − b − 1) n 3 F 2 −n, −x, n + a 1 + a 2 − b − 1 a 1 , a 2 ; 1 , from which (58) and (59) can be obtained using HolonomicFunctions. As we observed in [6], the finite family of polynomials that are usually called "Hahn polynomials" in the literature [33, 18.19]…”

Discrete semiclassical orthogonal polynomials of class one

“…The moments of discrete semiclassical orthogonal polynomials of class s ≤ 1 were studied in [18]. 1.…”

Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations

“…where (•) k is the usual Pochhammer symbol and δ m,n is the Kronecker delta. This collection of polynomials is known as the discrete orthogonal polynomials with hypergeometric weights, since the moments of this weight function are given in terms of the Gauss hypergeometric function 2 F 1 (α, β; γ; c) and its derivatives; it has been recently studied in [FVA18,Dom16]. These polynomials satisfy the three term recurrence relation…”