Generalized Factorial Functions and Binomial Coefficients
نویسنده
چکیده
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 )
منابع مشابه
Asymptotic Expansions of Gamma and Related Functions, Binomial Coefficients, Inequalities and Means
We give an overview of the use of asymptotic expansions of gamma and related functions — ratio of gamma functions, powers, digamma and polygamma functions. The aim is to introduce a general theory which can unify various particular formulas for factorial functions and binomial coefficients. The connection with inequalities for gamma function is established. Also, a systematic approach to asympt...
متن کاملCharacterization of the factorial functions of Eulerian binomial and Sheffer posets
We completely characterize the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or 2n−1, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n) = n! has...
متن کاملClassification of the factorial functions of Eulerian binomial and Sheffer posets
We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or 2n−1, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n...
متن کاملar X iv : m at h - ph / 0 40 10 06 v 1 5 J an 2 00 4 Calculation of some determinants using the s - shifted factorial
Abstract Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer’s symbol) and the falling factorial. It is a special ...
متن کاملar X iv : m at h - ph / 0 40 10 06 v 2 7 M ay 2 00 4 Calculation of some determinants using the s - shifted factorial
Abstract Several determinants with gamma functions as elements are evaluated. These kinds of determinants are encountered, for example, in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer’s symbol) and the falling factorial. ...
متن کامل