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Generalized ***j*-Factorial Functions, Polynomials, and Applications

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Maxie D. Schmidt

University of Illinois, Urbana-Champaign

Urbana, IL 61801

USA

**Abstract:**

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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