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Combinatorial Identities for Generalized Stirling Numbers Expanding
***f*-Factorial Functions and the *f*-Harmonic Numbers

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Maxie D. Schmidt

School of Mathematics

Georgia Institute of Technology

Atlanta, GA 30332

USA

**Abstract:**

We introduce a class of *f*(*t*)-factorials, or *f*(*t*)-Pochhammer symbols, that
includes many, if not most, well-known factorial and multiple factorial
function variants as special cases. We consider the combinatorial
properties of the corresponding generalized classes of Stirling numbers of
the first kind that arise as the coefficients of the symbolic polynomial
expansions of these *f*-factorial functions. The combinatorial properties
of these more general parameterized Stirling number triangles include
analogs of known expansions of the ordinary Stirling numbers by *p*-order
harmonic number sequences, through the definition of a corresponding
class of *p*-order *f*-harmonic numbers.

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(Concerned with sequences
A000142
A000165
A001008
A001147
A002805
A006882
A007318
A008275
A008517
A094638.)

Received March 29 2017; revised version received February 24 2018.
Published in *Journal of Integer Sequences*, March 7 2018.

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