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Eulerian numbers with fractional order parameters. (English) Zbl 0797.11025

Summary: The aim of this paper is to generalize the well-known “Eulerian numbers”, defined by the recursion relation \(E(n,k) = (k+1) E(n-1,k) + (n-k) E(n-1,k-1)\), to the case that \(n \in \mathbb{N}\) is replaced by \(\alpha \in \mathbb{R}\). It is shown that these “Eulerian functions” \(E (\alpha,k)\), which can also be defined in terms of a generating function, can be represented as a certain sum, as a determinant, or as a fractional Weyl integral. The \(E(\alpha,k)\) satisfy recursion formulae, they are monotone in \(k\) and, as functions of \(\alpha\), are arbitrarily often differentiable. Further, connections with the fractional Stirling numbers of second kind, the \(S(\alpha,k)\), \(\alpha>0\), introduced by the authors and M. Schmidt [Result. Math. 16, 16-48 (1989; Zbl 0707.05002)], are discussed. Finally, a certain counterpart of the famous Worpitzky formula is given; it is essentially an approximation of \(x^ \alpha\) in terms of a sum involving the \(E(\alpha,k)\) and a hypergeometric function.

MSC:

11B83 Special sequences and polynomials
26A33 Fractional derivatives and integrals
33C05 Classical hypergeometric functions, \({}_2F_1\)
11B68 Bernoulli and Euler numbers and polynomials
11B37 Recurrences
05A10 Factorials, binomial coefficients, combinatorial functions

Citations:

Zbl 0707.05002
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References:

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