Journal of Integer Sequences, Vol. 13 (2010), Article 10.6.7

Generalized j-Factorial Functions, Polynomials, and Applications

Maxie D. Schmidt
University of Illinois, Urbana-Champaign
Urbana, IL 61801


The paper generalizes the traditional single factorial function to integer-valued multiple factorial (j-factorial) forms. The generalized factorial functions are defined recursively as triangles of coefficients corresponding to the polynomial expansions of a subset of degenerate falling factorial functions. The resulting coefficient triangles are similar to the classical sets of Stirling numbers and satisfy many analogous finite-difference and enumerative properties as the well-known combinatorial triangles. The generalized triangles are also considered in terms of their relation to elementary symmetric polynomials and the resulting symmetric polynomial index transformations. The definition of the Stirling convolution polynomial sequence is generalized in order to enumerate the parametrized sets of j-factorial polynomials and to derive extended properties of the j-factorial function expansions. The generalized j-factorial polynomial sequences considered lead to applications expressing key forms of the j-factorial functions in terms of arbitrary partitions of the j-factorial function expansion triangle indices, including several identities related to the polynomial expansions of binomial coefficients. Additional applications include the formulation of closed-form identities and generating functions for the Stirling numbers of the first kind and r-order harmonic number sequences, as well as an extension of Stirling's approximation for the single factorial function to approximate the more general j-factorial function forms.

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(Concerned with sequences A000079 A000108 A000110 A000142 A000165 A000254 A000367 A000392 A000399 A000407 A000454 A000984 A001008 A001147 A001296 A001297 A001298 A001620 A001813 A002445 A002805 A006882 A007318 A007406 A007407 A007408 A007409 A007559 A007661 A007662 A007696 A008275 A008276 A008277 A008278 A008292 A008297 A008517 A008542 A008543 A008544 A008545 A008546 A008548 A008585 A027641 A027642 A032031 A034176 A045754 A045755 A047053 A048993 A048994 A052562 A066094 A080417 A081051 A094638 A098777 A111593 A130534 A154959.)

Received July 21 2009; revised version received June 19 2010. Published in Journal of Integer Sequences, June 21 2010.

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