The \(p\)-adic \(z\)-transform
type: Item
instance of: scholarly article
title: The \(p\)-adic \(z\)-transform
zbMATH Open document ID: 0844.11074
author: Lucien van Hamme
published in: Annales Mathématiques Blaise Pascal
publication date: 1996-09-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/79109
review text: Let \(a+ p^n \mathbb{Z}_p\) be a ball in \(\mathbb{Z}_p\). For a given \(z\in \mathbb{C}_p\) let \({\mathfrak m}_z (a+p^n \mathbb{Z}_p)= z^a/ (1- z^{p^n})\), where \(z\in \mathbb{C}_p\), \(|z-1 |_p\geq 1\). The \(p\)-adic distribution \({\mathfrak m}_z\) defines a kind of \(z\)-transform by \(F(z)= \int_{\mathbb{Z}_p} f(x) {\mathfrak m}_z (x)\). The author presents a systematic study of this transform. Moreover he gives some applications, e.g. to Mahler's theorem on representations of continued functions \(f: \mathbb{Z}_p\to \mathbb{Q}_p\) in the form \(f(x)= \sum^\infty_{n=0} a_n \binom x n\), van der Put expansions and to the expansion of a \(p\)-adic function in a series of Sheffer polynomials.
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 46S10
zbMATH DE Number: 800321
zbMATH Keywords: \(p\)-adic \(z\)-transform
zbMATH Keywords: \(p\)-adic distribution
zbMATH Keywords: representations of continued functions
zbMATH Keywords: van der Put expansions
zbMATH Keywords: Sheffer polynomials
reviewed by: Witold Więsław
MaRDI profile type: Publication
OpenAlex ID: W2094065813
cites work: Q3726220
cites work: Q3346515
cites work: Q3880931
cites work: Fonctions zeta p-adiques des corps de nombres abeliens réels
cites work: Q3203462
cites work: Q4016609
DOI: 10.5802/AMBP.25
description: scientific article; zbMATH DE number 800321