Journal of Integer Sequences, Vol. 3 (2000), Article 00.2.4

On Generalizations of the Stirling Number Triangles

Wolfdieter Lang
Institut für Theoretische Physik
Universität Karlsruhe
Kaiserstraße 12, D-76128 Karlsruhe, Germany
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Abstract: 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 xk d/dx, for k in Z.

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(Concerned with sequences A000007 A000012 A000045 A000079 A000085 A000108 A000110 A000142 A000262 A000369 A001147 A001497 A001515 A001700 A001710 A001715 A001720 A001725 A001792 A004747 A007318 A007559 A007696 A008275 A008277 A008279 A008297 A008543 A008544 A008545 A008546 A008548 A011801 A013988 A015735 A016036 A019590 A023531 A025748 A025749 A025750 A025751 A025756 A025757 A025758 A025759 A028575 A028844 A030523 A030524 A030526 A030527 A030528 A033184 A033842 A034171 A034255 A034687 A035323 A035324 A035342 A035469 A035529 A036068 A036070 A036083 A039717 A039746 A043553 A045624 A046088 A046089 A048882 A048965 A048966 A049027 A049028 A049029 A049118 A049119 A049120 A049213 A049223 A049224 A049323 A049324 A049325 A049326 A049327 A049348 A049349 A049350 A049351 A049353 A049374 A049375 A049376 A049377 A049378 A049385 A049402 A049403 A049404 A049410 A049411 A049412 A049424 A049425 A049426 A049427 A049431 A053113)

Received Feb. 11, 2000; published in Journal of Integer Sequences Sept. 13, 2000; minor editorial changes Nov. 30, 2000.

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