\(q\)-Taylor and interpolation series for Jackson \(q\)-difference operators
type: Item
instance of: scholarly article
title: \(q\)-Taylor and interpolation series for Jackson \(q\)-difference operators
zbMATH Open document ID: 1149.40001
author: Elsie Sterbin Gottlieb
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-05-28T00:00:00Z
review text: It is outside the scope of a review to give the formulae needed to understand the main results of the paper explicitly. For readers knowledgable in \(q\)-theory these main results will be stated below: \textbf{A.} Let \(0<R\leq \infty\) and \(f\) be analytic on \(\Omega_R\) with power series expansion \[ f(x)=\sum_{n=0}^{\infty}\,c_n x^n,\;x\in\Omega_R. \] Then \(f\) has the \(q\)-Taylor expansion \[ f(x)=\sum_{k=0}^{\infty}\,{D_q^k f(a)\over\Gamma_q(k+1)}\,\varphi_k(x,a), \] converging absolutely and uniformly on compact subsets of \(\Omega_R\). \textbf{B.} Let \(f(x)\) be a function with \(q\)-exponential growth of order \(k,\;k<\ln q^{-1}\), and finite type \(\alpha,\;\alpha\in\mathbb R\). Then for \(a\in\mathbb C\setminus\{0\}\), \(f(x)\) has the expansion \[ f(x)=\sum_{n=0}^{\infty}\,(-1)^n q^{-n(n-1)/2}\,{D_n^qf(aq^{-n})\over\Gamma_q(n+1)}\,\varphi_n(a,x), \] converging absolutely and uniformly on compact subsets of \(\mathbb C\).
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 40A30
Mathematics Subject Classification ID: 33D05
Mathematics Subject Classification ID: 39A70
Mathematics Subject Classification ID: 47B39
zbMATH DE Number: 5279807
zbMATH Keywords: \(q\)-Taylor series
zbMATH Keywords: Jackson \(q\)-difference operator
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.02.033
OpenAlex ID: W1981973883
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DOI: 10.1016/J.JMAA.2008.02.033
description: scientific article; zbMATH DE number 5279807