A kind of orthogonal polynomials and related identities

[摘要] In this paper we introduce the polynomials {d(n)((r)) (x)} and {D-n((r)) (x)} given by d(n)((r)) (x) = Sigma(n)(k=0) [GRAPHICS] [GRAPHICS] (n >= 0), D-0((r)) (x) = 1, D-1((r)) (x) = x and D-n+1((r)) (x) = xD(n)((r))(x) - n(n + 2r)D-n-1((r)) (x) (n >= 1). We show that {D-n((r)) (x))} are orthogonal polynomials for r > -1/2 and establish many identities for {d(n)((r)) (x)} and {D-n((r)) (x)}, especially obtain a formula for d(n)((r)) (x)(2) and the linearization formulas for d(m)((r)) (x)d(n)((r)) (x) and D-m((r)) (x)D-n((r)) (x). As an application we extend recent work of Sun and Guo. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Orthogonal polynomial;Identity;Three-term recurrence