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Abstract The theory of generalized hypergeometric functions is fundamental in the field of mathematical physics, since all the commonly used functions of analysis (Bessel Functions, Legendre Functions, etc.) are special cases of the general functions. The unified theory provides a means for the analysis of the simpler functions and can be used to solve the more complicated equations in physics. The study of hypergeometric single variable functions is over 200 years old. They come up in Euler, Gauss, Riemann, and Kummer’s research. Barnes and Mellin studied their integral representations, and Schwarz and Goursat studied special properties of them. The main developments until the end of the 1930ies were summarized by W.N. Bailey in the fundamental monograph [1]. Gauss hypergeometric function 2F1 and its confluent case 1F1 form the core special functions and include, as their special cases, most of the commonly used functions. Thus 2F1 includes, as its special cases, Legendre function, the incomplete beta function, the complete elliptic functions of first and second kinds, and most of the classical orthogonal polynomials, [2]. On the other hand, the confluent hypergeometric function includes, as its special cases, Bessel functions, parabolic cylindrical functions, and Coulomb wave function. The various interpretations of Gauss’ hypergeometric function have challengedmathematicians to generalize this function. This new-found interest comes from the connections between hypergeometric functions and many areas of mathematics such as representation theory, algebraic geometry and combinatorics, D-modules, number theory, etc. In the end of the 19th century and the beginning of the 20th century hypergeometric functions in several variables were introduced. For example Appells functions, the Lauricella functions and the Horn series. These types of series appear very naturally in quantum field theory, in particular in the computation of analytic expressions for Feynman integrals. Basic hypergeometric series have assumed great importance during the last four decades or so because of their applications in diverse fields, like additive number theory, combinatorial analysis, statistical and quantum mechanics, vector spaces etc, [3]. A fresh interest in these functions was aroused by the discovery of Ramanujan’s ”Lost” Note book by G.E. Andrews in 1976, [4]. A beautiful account of the discovery of the ”Lost” Notebook and its contents, has been given by him in 1979 in the American Mathematical Monthly. The enormous mass of literature on basic hypergeometric series (or q-hypergeometric series as we often call it) has become so significant and important that their study has acquired an independent, respectable status of its own rather merely being treated as a generalization of the ordinary hypergeometric series, [5, 6]. The main aims of this thesis are to introduce: 1. A study of classical summation theorems for the series 2F1 and 3F2 and their various applications obtained by earlier researchers and to extend our study to the problems of their generalizations and extensions in order to obtain generalized extensions of such theorems as well as obtaining new hypergeometric identities and summations. 2. Various computations concerning the contiguous function relations of the hypergeometric and basic hypergeometric functions. 3. A study of using such summation theorems in order to compute integrals involving generalized hypergeometric function. 4. A study of how we can obtain unknown Laplace transforms of generalized hypergeometric function 2F2[a; b; c; d; x] by employing generalizations of Gauss’ second, Bailey’s and Kummer’s summation theorems. 5. A study of the q-analogues of some of the classical summation theorems and their q-extensions. The thesis consists of Four main chapters: Chapter 1: This chapter contains an introduction to hypergeometric series, hypergeometric functions, generalized hypergeometric functions and basic hypergeometric series(q-series). Chapter 2: deals with the integrals involving generalized and product of two generalized hypergeometric functions as well as investigate a new representation of generalized hypergeometric function 2F2 [a; b; c; d; x] by using Laplace transform. Chapter 3: In this chapter, we discuss extensions of some classical summation theorems such as Gauss theorem, Kummer theorem, Kummer’s second theorem and others, also we evaluate some reduction formulas for hypergeometric functions of two or more variables. Chapter 4: The main goal of this chapter is to obtain new q-analogue and q-contiguous formulas of some well-known classical summation theorems. |