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Factorial functions and stirling numbers of fractional orders

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Abstract

The aim of this paper is to study the binomial coefficients ( xn ), the factorial polynomials [x]n and [x]n, the Stirling numbers of first and second kind, namely s(n,k) and S(n,k), in the case that n ∈ ℕ is replaced by real α ∈ ℝ. In the course of the paper, the Vandermonde convolution formula is presented in an infinite series frame, the binomial coefficient function ( xa ), α ∈ ℝ, is sampled in terms of the binomial coefficients ( xk ) for k ∈ ℕo, Bell numbers of fractional orders are introduced. Emphasis is placed on the fractional order Stirling numbers s(α,k) and S(α,k), first studied here. Some applications of the S(α,k) are given.

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  1. Lehrstuhl A für Mathematik, Rheinisch-Westfälische Technische Hochschule Aachen, Templergraben 55, D-5100, Aachen, Federal Republic of Germany

    P. L. Butzer, M. Hauss & M. Schmidt

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  1. P. L. Butzer
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  2. M. Hauss
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  3. M. Schmidt
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Butzer, P.L., Hauss, M. & Schmidt, M. Factorial functions and stirling numbers of fractional orders. Results. Math. 16, 16–48 (1989). https://doi.org/10.1007/BF03322642

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