On moments of classical orthogonal polynomials

Abstract

The aim of this work is to find simple formulas for the moments µ n for all families of classical orthogonal polynomials listed in the book by Koekoek, Lesky and Swarttouw [30]. The generating functions or exponential generating functions for those moments are given.To my dear parents

Evaluations of certain Catalan-Hankel Pfaffians via classical skew orthogonal polynomials

“…With a = −1, Al-Salam & Carlitz I polynomials reduce to the q-Hermite I polynomials [22, section 14.28], and therefore the above mentioned method could be applied to the q-Hermite I case as well. The moment of q-Hermite I polynomials takes the form [27] μ n = (1 − q)(q, −1, −q; q) ∞ 1 + (−1) n 2 q; q 2 n/2 , and we have following evaluation of q-Catalan-Hankel Pfaffian Pf (q i − q j ) 1 + (−1) i+ j−1 2 (q; q 2 ) i+ j−1…”

“…In this case, canonical moments are given by [27] μ (α,β) n = 1 0 z α+n (qz; q) β d q z = (1 − q)(q α+β+2 ; q) ∞ (q; q) ∞ (q α+1 ; q) n (q α+1 ; q) ∞ (q β+1 ; q) ∞ (q α+β+2 ; q) n .…”

“…) n q ( n 2 ) (q, a −1 c, ac −1 q, abq; q) ∞ (q, aq, bq, cq, abc −1 q; q) n (1 − abq 2n+1 )(aq, bq, cq, abc −1 q; q) ∞ (abq; q) n (abq n+1 ; q) 2 n and the canonical moments are given by [27] μ (a,b,c) n = aq cq x n ρ (a,b,c) (x; q)d q x = aq(1 − q) q, abq 2 , a −1 c, ac −1 q; q ∞ aq, bq, cq, abc −1 q; q ∞ × n m=0 (−1) m n m q q −nm+ m+1 2 (aq, cq; q) m abq 2 ; q m .…”

Polynomial Sequences Associated with the Moments of Hypergeometric Weights

“…In [28], the authors used (11) The moments of these polynomials have not (to our knowledge) been studied before.…”

“…Given their importance, it is very striking that they are not explicitly listed in the standard books on orthogonal polynomials, or even in encyclopedic texts such as [23]. In fact, the only place where we found a comprehensive enumeration of the moments of classical orthogonal polynomials was the recent article [28], based on the results obtained in the Ph.D. Thesis of the first author [27] 1 .…”