A \(q\)-analog of Newton's series, Stirling functions and Eulerian functions
type: Item
instance of: scholarly article
title: A \(q\)-analog of Newton's series, Stirling functions and Eulerian functions
zbMATH Open document ID: 0816.33010
DOI: 10.1007/BF03323418
author: Changgui Zhang
author: Jiang Zeng
published in: Results in Mathematics
publication date: 1995-07-13T00:00:00Z
review text: In a series of papers, \textit{P. L. Butzer}, \textit{M. Haus} and \textit{M. Schmidt} [Result Math. 16, No. 1/2, 16-48 (1989; Zbl 0707.05002); \textit{P. L. Butzer} and \textit{M. Haus}, Stud. Appl. Math. 84, No. 1, 71-91 (1991; Zbl 0738.11025); \textit{M. Haus} and \textit{M. Schmidt}, Aequationes Math. 46, No. 1/2, 119-142 (1993; Zbl 0797.11025)] studied extensions to complex arguments of classical combinatorial functions, such as Stirling numbers and Eulerian numbers. In the paper under review, the project of doing the same with the corresponding \(q\)-analogues is started. For \(\alpha\in \mathbb{C}\) and \(n\in \mathbb{N}\) the authors introduce the \(q\)-Stirling function \(S_ q(\alpha, n)\) of the second kind and the \(q\)-Eulerian function \(A_ q(\alpha, n)\), which for \(\alpha\in \mathbb{N}\) agree with the usual \(q\)-Stirling numbers of the second kind and the usual \(q\)-Eulering numbers, respectively. The authors derive several properties and prove various identities for these functions. An example is the identity \(\Gamma_ q(x+ 1)= \sum_{s\geq 0} A_ q(x,s)\), provided \(\text{Re } x>-1\), where \(\Gamma_ q(z)\) is the usual \(q\)-gamma function.
Mathematics Subject Classification ID: 33D05
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 33B15
zbMATH DE Number: 604861
reviewed by: C. Krattenthaler
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03323418
OpenAlex ID: W2038731483
description: scientific article; zbMATH DE number 604861