Double interpolation of the Euler numbers
The Eulerian numbers, which have well known combinatorial properties, are defined as \[ A(k, n)= \sum_{p=0}^{k-1} (-1)^p \binom{n+1}{p} (k-p)^n \] for \(1\le k\le n\), \(k\), \(n\), integers. The paper gives two methods for interpolating \(A\) by a function of two real variables \(A(x, y)\) (with \(y>-1\)), such that the following properties are retained: \[ A(x, y+1)= (y+2-x) A(x-1, y)+ xA(x, y), \qquad A(x,y)= A(y+1-x,y). \] The first formula for \(A\) is an explicit piecewise polynomial, the second formula is an integral expression, and the main result of the paper is that they give the same function. Some further properties of the function \(A\) are obtained in the paper, among which the computation of the integral \(\int_{-\infty} ^{+\infty} A(x,y) \,dx= \Gamma (y+1)\) which generalizes Worpitzky's formula \(\sum_{k=1}^ n A(k, n)= n!\).
scientific article; zbMATH DE number 731894
scientific article