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Factorial functions and Stirling numbers of fractional orders. (English) Zbl 0707.05002
It has long been known that its expression in terms of factorials enables the binomial coefficient \(\left( \begin{matrix} x\\ \alpha \end{matrix} \right)\) to be extrapolated to complex values of x and \(\alpha\) by means of the \(\Gamma\)-function. The authors give the extended forms of identities such as Vandermonde’s convolution formula, valid for x in the half-plane \(Re(x)>-1,\) which they show is equivalent to Gauss’s Summation Theorem expressing the hypergeometric function in terms of a series involving rising factorials. A new feature is that, unlike the \(\Gamma\)-function, \(\left( \begin{matrix} x\\ \alpha \end{matrix} \right)\) is band-limited as a function of \(\alpha\) for \(Re(x)>-1,\) so in this range the extended values of \(\left( \begin{matrix} x\\ \alpha \end{matrix} \right)\) can be obtained by Nyquist-Shannon interpolation from the values for integer \(\alpha\). Indeed, the interpolation formula is a special case of the convolution formula. Also the Stirling numbers, s(\(\alpha\),k) and S(\(\alpha\),k), being the coefficients in the expression of the falling factorials \([x]_{\alpha}\quad (=\Gamma (x+1)/\Gamma (x-\alpha +1))\) as a sum of powers \(x^ k\) and vice versa, can be extrapolated to complex values of \(\alpha\). It is shown that \(s(\alpha,k)/\Gamma (\alpha +1)\) is given by Nyquist-Shannon interpolation of its values for integer \(\alpha\), and other identities are proved. The Bell numbers B(\(\alpha\)) are also extrapolated to complex \(\alpha\), both via their expression as the sum of the S(\(\alpha\),k) and, equivalently, via the Dobinski formula.
Reviewer: P.A.B.Pleasants

MSC:
05A10 Factorials, binomial coefficients, combinatorial functions
05A19 Combinatorial identities, bijective combinatorics
11B65 Binomial coefficients; factorials; \(q\)-identities
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
33B15 Gamma, beta and polygamma functions
33C99 Hypergeometric functions
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