A \(q\)-analog of Newton's series, Stirling functions and Eulerian functions
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.
scientific article; zbMATH DE number 604861