A \(q\)-analogue of Mahler expansions. I
type: Item
instance of: scholarly article
title: A \(q\)-analogue of Mahler expansions. I
zbMATH Open document ID: 1003.11055
author: Keith Conrad
published in: Advances in Mathematics
publication date: 2001-01-30T00:00:00Z
review text: The author studies a \(q\)-analogue of Mahler expansions for continuous functions \(f:\;\mathbb Z_p\to \mathbb Q_p\). The binomial coefficient polynomials \(\binom xn\) are replaced with their \(q\)-analogue \(\binom xn_q\), \(q\in \mathbb Z_p\), \(|q-1|_p<1\). Four different proofs of the expansion theorem are given. A formula for the coefficients in the expansion of \(f\) is obtained, as well as characterizations of differentiability and analyticity in terms of the above coefficients, and a formula for the expansion into a finite part of the \(\binom xn_q\)-series and a remainder term. An interpretation of the \(p\)-adic interpolation of Koblitz's \(p\)-adic \(q\)-Gamma function in terms of its \(q\)-Mahler expansion is proposed. Earlier \(\binom xn_q\)-expansions were obtained for functions defined on certain closed subsets of \(\mathbb Z_p\); see \textit{Y. Amice} [Bull. Soc. Math. Fr. 92, 117--180 (1964; Zbl 0158.30203)], \textit{L. van Hamme} [Proc. Conf. on \(p\)-adic Analysis, Nijmegen, 119--125 (1978)], and \textit{A. Verdoodt} [Indag. Math., New Ser. 4, 375--384 (1993; Zbl 0788.11056)].
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 12J25
zbMATH DE Number: 1507786
zbMATH Keywords: Mahler expansion
zbMATH Keywords: \(q\)-binomial coefficients
zbMATH Keywords: \(p\)-adic \(q\)-Gamma function
reviewed by: Anatoly N. Kochubei
MaRDI profile type: Publication
full work available at URL: aima.1999.1890
OpenAlex ID: W2070904695
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DOI: 10.1006/AIMA.1999.1890
description: scientific article; zbMATH DE number 1507786