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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.

MSC:
 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
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References:
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