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 and rings of integer-valued polynomials. In Chapter 2, necessary conditions on an infinite sequence of integers are obtained in order for that sequence to serve as the factorial sequence for some subset S ⊆ Z. Chapter 3 explores the subject of !-equivalent subsets and we find a condition on two infinite subsets S and T of Z which force n!S = n!T for every nonnegative integer n. We close in Chapter 4 with an analysis of generalized binomial coefficients, and for a given infinite subset S ⊆ Z, we characterize those subsets T ⊆ Z for which ( n m )