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On Stirling functions of the second kind. (English) Zbl 0738.11025
The authors prove several results for Stirling functions defined by \(S(\alpha,k)=(1/k!)\Delta^ kx^ \alpha|_{x=0},\) \(\alpha\geq 0\); \(k\in{\mathbb{N}}_ 0\), viewed as function of \(\alpha\), where \(\Delta\) is the forward difference operator. Among the results obtained are proofs of the continuity and differentiability of \(S\), recurrence relations, real integral representations, a representation in terms of the Weyl derivative of fractional order \(\alpha\), and connections with the Bernoulli, Stirling, and Bernstein polynomials and with Bernoulli numbers of fractional order.

MSC:
11B73 Bell and Stirling numbers
11B68 Bernoulli and Euler numbers and polynomials
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