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A \(q\)-analog of Newton’s series, Stirling functions and Eulerian functions. (English) Zbl 0816.33010
In a series of papers, P. L. Butzer, M. Haus and M. Schmidt [Result Math. 16, No. 1/2, 16-48 (1989; Zbl 0707.05002); P. L. Butzer and M. Haus, Stud. Appl. Math. 84, No. 1, 71-91 (1991; Zbl 0738.11025); M. Haus and 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.

33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
05A10 Factorials, binomial coefficients, combinatorial functions
05A19 Combinatorial identities, bijective combinatorics
11B65 Binomial coefficients; factorials; \(q\)-identities
11B73 Bell and Stirling numbers
33B15 Gamma, beta and polygamma functions
Full Text: DOI
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