Q-Combinatorial Identities

The basic hypergeometric series was studied essentially started in 1748 by Euler. But after a hundred years the subject acquired an independent status when Heine observed ₂φ₁ which is parrel to Gauss's ₂F₁ hypergeometric series. In view of the base q, the basic hypergeometric series is also called q-series, or q-hypergeometric series. During 19-20 century, L.J. Rogers, F.H. Jackson, A.C. Dixon, J. Dougall, L. Saalschutz, F.J.W. Whipple, G.N. Watson, W.N. Bailey, J.A. Daum and other mathematicians derived many important results on basic hypergeometric series. In these years, basic hypergeometric series was started to explore their applications in some areas of pure and applied mathematics and other subjects. G.E. Andrews and N. J. Fine gave some application of basic hypergeometric series to number theory, association schemes, combinatorics, difference equation, Lie algebra, physics, statistics, etc.The key point of this thesis is a part of the progress in basic hypergeometric series including the q-positive integers, q-dual sequence and their applications. In Chapter 1. the history of hypergeometric; series and basic hypergeometric series are introduced. To make the following topics easier understandable, meanwhile, some basic definitions of classical notations, basic identities, and transformations are also listed in the first chapter.At the beginning of Chapter 2, we present a new family of positive integers which was derived by M. Lassalle. g-Positive integers is defined as the coefficient of a q-polynomials is non-negative. Then by using q-binomial theorem and Sears's transformation formula for ₃φ₂, we show two new families of g-positive integers, which are further extended to a special case of the coefficients of the product of q-binomial polynomials.Dual sequences with applications were researched by Zhi-Wei Sun. The new recursion formula and symmetric extension about Bernoulli numbers and Bernoulli polynomials were also be published by Kaneko, Momiyama and Wu et al. In chapter 3, we introduce g-dual sequences and their applications in basic hypergeometric series, especially about g-Bernoulli numbers and g-Bernoulli polynomials which were defined by Carlitz and Al-Salam.