Geodesic
Generating functions for extended Stirling numbers of the first kind
Journal of integer sequences, Tome 17 (2014) no. 6
Zbl
In this paper we extend the definition of Stirling numbers of the first kind by way of a special multiset. This results in a family of number triangles for which we show how to obtain ordinary generating functions for the rows and exponential generating functions for the columns. The latter are derived via a recursive process. We also indicate how to obtain formulas, in terms of factorials, generalized harmonic numbers, and polynomials, for the entries in the columns of these number triangles.
Classification : 11B73, 05A05, 05A10, 05A15, 11B37
Keywords: Stirling number of the first kind, multiset, permutation, cycle, recurrence relation, generating function 
Griffiths,  Martin. Generating functions for extended Stirling numbers of the first kind. Journal of integer sequences, Tome 17 (2014) no. 6. http://geodesic.mathdoc.fr/item/JIS_2014__17_6_a0/
@article{JIS_2014__17_6_a0,
     author = {Griffiths,  Martin},
     title = {Generating functions for extended {Stirling} numbers of the first kind},
     journal = {Journal of integer sequences},
     year = {2014},
     volume = {17},
     number = {6},
     zbl = {1358.11042},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2014__17_6_a0/}
}
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