The symmetric Meixner-Pollaczek polynomials with real parameter
type: Item
instance of: scholarly article
title: The symmetric Meixner-Pollaczek polynomials with real parameter
zbMATH Open document ID: 1077.33009
author: Tsehaye K. Araaya
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-28T00:00:00Z
review text: The symmetric Meixner-Pollaczek polynomials \(p_n^\lambda(x)\) can be determinated by the generating function \[ {{\exp (x\arctan t)}\over {(1+t^2)^\lambda}}=\sum_{n=0}^\infty p_n^\lambda(x) t^n,\quad \lambda\in\mathcal R. \] It is well known that these polynomials are orthogonal with the weight \[ \omega_\lambda(x)= { 1\over {2\pi}} \biggl| \Gamma \biggl( \lambda+ {{ix}\over 2}\biggr)\biggr| ^2 \] if \(\lambda>0\). In the case where \(\lambda\leq 0\) the Favard Theorem implies that there is no positive real measure \(\mu\) such that \(p_n^\lambda(x)\) are orthogonal with respect to \(\mu\). The author succeeded in proving that \(\{ p_n^\lambda(x)\}^\infty_{n=0}\) is an orthogonal polynomial system for every real \(\lambda\) if we define the associated inner product as follows: \[ \langle f, g\rangle_\lambda=\int_{-\infty}^{+\infty} R^{m_\lambda} (f\cdot g^*)\omega_{\lambda+{{m_\lambda}\over 2}} (x) \,dx \] where \(m_\lambda=\min_{n\in{\mathcal N}}\{ n: \lambda+ {n\over 2} >0\}\), \(R(f(x))={{f(x+i)+f(x-i)}\over 2}\), \(R^n(f)=R(R^{n-1}(f))\), \(g^*\equiv\) the complex conjugate of \(g\) and \(\omega_{\lambda+{{m_\lambda}\over 2}} (x)={1\over{2\pi}} | \Gamma(\lambda+{{m_\lambda}\over 2}+{{ix}\over 2})| ^2\).
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 2162010
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Meixner-Pollaczek polynomials
zbMATH Keywords: non-standard inner product
zbMATH Keywords: polynomial operators
reviewed by: Aleksey A. Dovgoshey
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.10.067
OpenAlex ID: W2035032140
cites work: The Meixner--Pollaczek polynomials and a system of orthogonal polynomials in a strip
cites work: Q4665077
cites work: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials
cites work: The Hahn and Meixner polynomials of an imaginary argument and some of their applications
cites work: Continuous Hahn polynomials and the Heisenberg algebra
cites work: Q4170993
cites work: Q4400929
cites work: A selected survey of umbral calculus
cites work: A generalization of Favard's theorem for polynomials satisfying a recurrence relation
cites work: Q4304847
cites work: Meixner–Pollaczek polynomials and the Heisenberg algebra
cites work: On the asymptotics of the Meixner-Pollaczek polynomials and their zeros
cites work: Q4376453
cites work: Q5798212
cites work: A Generalization of Gasper's Kernel for Hahn Polynomials: Application to Pollaczek Polynomials
cites work: The umbral calculus
DOI: 10.1016/J.JMAA.2004.10.067
description: scientific article; zbMATH DE number 2162010