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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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