Extension of Some Special Functions and Hypergeometric Functions

Abstract

A mathematical theory on the classical beta function B(and#945;, and#946;) is one of the most fundamental theory of special functions, which is used to solve theoretical as well as practical problems in various field of sciences such as pure and applied mathematics, physics, statistical sciences and engineering. Also, the term quotHypergeometric seriesquot was first used by John Wallis in his book quotArithmetica Infinitorumquot in 1655. Hypergeometric series is used first time systematically by Carl Fridrich Gauss in 1813. Hypergeometric function is a special function which represent hypergeometric series including many other special functions. newlineThe Mittag-Leffler functions arise in the solutions of fractional order integral equations and used in the investigation of the some other fractional or generalization form of extended special functions. newlineIn this thesis generalization in the form of extension of special functions has been made with their properties and various applications. The new extension of classical beta function is analyzed on the basis of numerical results, which is established by using MATLAB software. newlineIn this contribution, generalization in the form of extension of special functions has been made. The properties and applications are also given. The work is listed in chapter wise as: newline1. Chapter 1: This Chapter is an introductory in nature and provides say a brief introduction of various topics to study in all research work. newline2. Chapter 2: In this Chapter the available literature is discussed. A lot of work is published by various workers in the field of fractional calculus, generating functions, hypergeometric function etc. newline3. Chapter 3: In this Chapter, new type extension of classical beta function is introduced and proved its convergence. Further it is used to introduce the extension of Gauss hypergeometric function and confluent hypergeometric functions. Then, their properties, integral representation, certain fractional derivatives, fractional integral formulas and application of these functions has been studied.