Jocelyn Quaintance

Combinatorial identities for Stirling numbers : the unpublished notes of H.W. Gould

This book is a unique work which provides an in-depth exploration into the mathematical expertise, philosophy, and knowledge of H W Gould. It is written in a style that is accessible to the reader with basic mathematical knowledge, and yet contains material that will be of interest to the specialist in enumerative combinatorics. This book begins with exposition on the combinatorial and algebraic techniques that Professor Gould uses for proving binomial identities. These techniques are then applied to develop formulas which relate Stirling numbers of the second kind to Stirling numbers of the first kind. Professor Goulds techniques also provide connections between both types of Stirling numbers and Bernoulli numbers. Professor Gould believes his research success comes from his intuition on how to discover combinatorial identities.This book will appeal to a wide audience and may be used either as lecture notes for a beginning graduate level combinatorics class, or as a research supplement for the specialist in enumerative combinatorics.

Double Fun with Double Factorials

Summary The double factorial of n may be defined inductively by (n + 2)!! = (n + 2)(n)!! with (0)!! = (1)!! = 1. Alternatively we may define this notion by the two relations (2n)!! = 2 ·4 · 6 · 8…(2n) =2nn! and (2n - 1)!! = 1 · 3 · 5 · 7…(2n - 1) = (2n)!/2n!. Our object is to exhibit some properties and identities for the double factorials. Furthermore, we extend the notion of double factorial to the binomial coefficients by introducing double factorial binomial coefficients. The double factorial binomial coefficient is defined as We derive identities and generating functions involving these double factorial binomial coefficients.

Bell Numbers and Variant Sequences Derived from a General Functional Differential Equation

Abstract It is well known that the Bell numbers have exponential generating function , which satisfies the differential equation . In this paper, we investigate certain sequences whose exponential generating functions satisfy a modified form of the above differential equation, namely, the functional differential equation . For the main result of this paper, we show that when a = –1 and b ∈ ℝ, the sequence obeys the simple second-order linear recurrence G(n + 2) = bG(n + 1) – G(n). The proof is based on a well-known binomial series inversion formula.