A Class Of Extended Hypergeometric Functions And Their Applications

Recently,there emerge many new extensions of the beta functions and the hypergeomet-ric functions.The constructions of these extended beta functions mainly depend on inserting some specific integral kernels in their integral representations.By means of the new extended beta function,a corresponding extension of hypergeometric function can be defined.Gaining enlightenment from these ideas,we attempt to construct a new class of extended hypergeo-metric functions such that many known functions can be regarded as some special cases of our functions.In this thesis,we will first introduce a new extension of Gauss hypergeomet-ric function.Many fundamental theorems including its Euler-type integral representation,derivative formulas,Mellin-Barnes integral representation etc.will be established.For some special cases,some nontrivial results such as the generalized Euler transformation,summa-tion formulas and its H-function expression will also be considered.Next,as a new extension of generalized hypergeometric function,we define the extended generalized hypergeometric functions.The importance of these functions are depicted by several noticeable identities and their interconnection with a class of fractional operators.Furthermore,by adopting this methodology at various stages,some extensions of Appell's and Lauricella's hypergeometric functions are also obtained.In addition to their basic properties,a set of generating functions are established by applying the fractional integral operators.