On generalizations of the Stirling number triangles

Lang, Wolfdieter

Journal of integer sequences, Tome 3 (2000) no. 2.

Résumé

Summary: Sequences of generalized Stirling numbers of both kinds are introduced. These sequences of triangles (i.e. infinite-dimensional lower triangular matrices) of numbers will be denoted by $S2(k;n,m)$ and $S1(k;n,m)$ with k in Z. The original Stirling number triangles of the second and first kind arise when k = 1. $S2(2;n,m)$ is identical with the unsigned $S1(2;n,m)$ triangle, called $S1p(2;n,m)$, which also represents the triangle of signless Lah numbers. Certain associated number triangles, denoted by $s2(k;n,m)$ and $s1(k;n,m)$, are also defined. Both $s2(2;n,m)$ and $s1(2;n + 1, m + 1)$ form Pascal's triangle, and $s2(-1,n,m)$ turns out to be Catalan's triangle. Generating functions are given for the columns of these triangles. Each $S2(k)$ and $S1(k)$ matrix is an example of a Jabotinsky matrix. Therefore the generating functions for the rows of these triangular arrays constitute exponential convolution polynomials. The sequences of the row sums of these triangles are also considered. These triangles are related to the problem of obtaining finite transformations from infinitesimal ones generated by $x^{k}$ d/dx, for k in Z.