الفهرس 
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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.