Generalized j-Factorial Functions, Polynomials, and Applications

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. 1 Notational Conventions Donald E. Knuth’s article Two Notes on Notation [33] establishes several of the forms of standardized notation employed in the article. In particular Knuth’s notation for the Stirling 1 number triangles and the Stirling polynomial sequences [24; 32] are used to denote these forms and reasonable extensions of these conventions are used to denote the generalizations of the forms established by this article. The usage of notation for standard mathematical functions is explained inline in the text where the context of the relevant forms apply [46]. The following is a list of the other main notational conventions employed throughout the article. • Indexing Sets: The natural numbers are denoted by the set notation N and are equivalent to the set of non-negative integers, where the set of integers is denoted by the similar blackboard set notation for Z. The standard set notation for the real numbers (R) and complex numbers (C) is used as well to denote scalar and approximate constant values. • Natural Logarithm Functions: The natural logarithm function is denoted Log(z) in place of ln(z) in the series expansion properties involving the function. Similarly Log(z) denotes the natural logarithm function raised to the k power. • Iverson’s Convention: The notation [condition]δ for a boolean-valued input condition represents the value 1 (or 0) where the input condition evaluates to True (or False). Iverson’s convention is used extensively in the Concrete Mathematics reference and is a comparable replacement for Kronecker’s delta function for multiple pairs of arguments. For example, the notation [n = k]δ is equivalent to δn,k and the notation [n = k = 0]δ is equivalent to δn,0δk,0. • Sequence Enumeration and Coefficient Extraction: The notation 〈gn〉 7→ {g0, g1, g2, . . .} denotes a sequence indexed over the natural numbers. Given the generating function F (z) representing the formal power series (also generating series expansion) that enumerates 〈fn〉, the notation [z]F (z) := fn denotes the series coefficients indexed by n ∈ N. • Fixed Parameter Variables: For an indexing variable n, the notation Nc is employed to represent a fixed parameter in a formula or generating function that is treated as a constant and that is only assigned the explicit value of the respective non-constant indexing variable after all other variables and indices have been input and processed symbolically in a relevant form. In particular the fixed Nc variable should be treated as a constant parameter in series or generating function closed-forms, even when the non-constant form of n refers to a particular coefficient index in the series expansion. The footnote on p. 18 clarifies the context and particular utility of the fixed parameter usage in a specific example inline in the text. 2 Introduction The parametrized multifactorial (j-factorial) functions studied in this article generalize the standard classical single factorial [A000142] and double factorial [A001147; A000165; and A006882] functions and are characterized by the analogous recursive property in (2.1). n!(j) := n (n− j)!(j) [n ≥ j]δ + [0 ≤ n < j]δ (2.1)