Integral transforms and Chebyshev-like polynomials.
type: Item
instance of: scholarly article
title: Integral transforms and Chebyshev-like polynomials.
zbMATH Open document ID: 1033.33004
DOI: 10.1016/S0096-3003(02)00839-1
published in: Applied Mathematics and Computation
publication date: 2003-12-03T00:00:00Z
review text: The author considers the Chebyshev polynomials \[ U_n(x, y)=\sum_{k=0}^{[n/2]}\frac{(n-k)! y^k x^{n-2k}}{k!(n-2k)!} \] and their \(m\)th-order generalized forms \[ U_n^{(m)}(x, y)=\sum_{k=0}^{[n/m]}\frac{(n-k)! y^k x^{n-mk}}{k!(n-mk)!} \] Some properties of these polynomials are obtained by means of the integral representation method. For instance, it was shown that the equation \[ \left\lfloor (4y+x^2)\;\frac{\partial^2}{\partial x^2} + 3x\frac{\partial}{\partial x} -n(n+2)\right\rfloor U_n(x, y)=0, \] and the recurrence \[ U_{n+1}^{(m)} (x, y)=x U_n^{(m)}(x, y) +(m-1)y \frac{\partial^{m-2}}{\partial x^{m-2}} U_{n-1}^{(m)} (x, y) \] hold. Other families of polynomials are also considered.
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 2012690
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: Humbert differential operators
author: G. Dattoli
reviewed by: Aleksey A. Dovgoshey
MaRDI profile type: Publication
cites work: Generalized polynomials and associated operational identities
cites work: Q5342658
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cites work: On a new family of Hermite polynomials associated to parabolic cylinder functions
description: scientific article; zbMATH DE number 2012690