Generalized $j$-factorial functions, polynomials, and applications

Schmidt, Maxie D.

Journal of integer sequences, Tome 13 (2010) no. 6.

Résumé

Summary: 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.

Détail

Keywords: multifactorial, factorial, j-factorial, double factorial, factorial polynomial, Stirling number, Stirling triangle, Stirling number of the first kind, Stirling polynomial, Pochhammer symbol, concrete mathematics, Bernoulli polynomial, Bernoulli number, N$\ddot $orlund polynomial, triangular recurrence, triangle, harmonic number, harmonic series, symmetric polynomial, binomial theorem, binomial coefficient, Newton series, Stirling's approximation, Stirling's formula, methodus differentialis, arithmetic progression, root of unity, exponential generating function, double generating function, super generating function, gamma function, polygamma function, exponential integral, modified Bessel function, lerch transcendent, Euler-mascheroni constant, Euler gamma, Eulerian number, Eulerian triangle, Wilson's theorem, Laguerre polynomial, umbral calculus, polynomial sequence, Appell sequence, Euler polynomial, Hermite polynomial