Study of techniques of evaluation and interpretation of special functions and integral transforms

Abstract

We have studied single and double Integration formula of single and multivariable newlineSpecial functions, Integral operators. Special Functions are particular newlinemathematical functions which have more or less established names and notations newlinedue to their importance in mathematical analysis functional analysis physics or newlineother applications. The lists of mathematical functions which are commonly newlineaccepted as special are called special function. In particular, elementary functions newlineBeta and Gamma functions are also considered as special functions. newlineUnderstanding of definitions and use of fractional calculus will be made clearer newlineby discussing some necessary but relatively simple mathematical definitions that newlinewill arise in the study of these concepts. newlinewe discuss Special Functions Integral Formula Integral transform newlineFractional Calculus and Integral operators.We have presented introduction and newlinereview of history of various special functions like Beta Function Gamma newlineFunction Gauss Hypergeometric function Bessel function Confluent newlineHypergeometric Kummer Function Generalized Hypergeometric Functions newlineMittag Leffler Function K Struve function K- Mittag Leffler Function newlineWrights generalized hyper geometric function and generalized special function newlineG- special function Pathway Integral Operator Oberhettinger integral formula newlineLavoi Trottier integral formula and Mac Robert integral formula. newlinewe discus with general theorems on composition of Bessel Functions newlineof first kind positive and negative order and modified Bessel s function of first newlinekind. newlinewe find some unified integrals involving families of generalized newlineMittag Leffler type functions. Some interesting integral transforms of these newlineintegrals are also presented here as particular cases. newlinewe find some new results for Pathway integral operator, including the newlinek Struve Function and (p q) Extended Struve Function as the kernel newlinefactor. newlinepresents certain unified integrals define by Edward and Lavoie- newlineTrottier involving generalized function. Composition of these results newlineare further expressed