Generalized factorial series

Discrete Gamma (Factorial) Function and Its Series in Terms of a Generalized Difference Operator

On the irrationality of factorial series II

In this paper we give irrationality results for numbers of the form ∑∞ n=1 an n! where the numbers an behave like a geometric progression for a while. The method is elementary, not using differentiation or integration. In particular, we derive elementary proofs of the irrationality of π and em for Gaussian integers m 6= 0.

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 number...

Bernoulli numbers and generalized factorial sums

We prove a pair of identities expressing Bernoulli numbers and Bernoulli numbers of the second kind as sums of generalized falling factorials. These are derived from an expression for the Mahler coefficients of degenerate Bernoulli numbers. As corollaries several unusual identities and congruences are derived.

Generalized Factorial Functions and Binomial Coefficients

Let S ⊆ Z. The generalized factorial function for S, denoted n!S , is introduced in accordance with theory already established by Bhargava ([4]). Along with several known theorems about these functions, a number of other issues will be explored. This Thesis is divided into 4 chapters. Chapter 1 provides the necessary definitions and offers a connection between the generalized factorial function...