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ABSTRACT CHAPTER 1 A brief survey of the work done on fractional calculus andgeneralized integral transformation CHAPTER 2 (1) The relation between double Laplace transform and double Hankel transform with applications (2) Relationship between double Laplace transform and double Mellin transform in terms of generalized hypergeometric function with applications CHAPTER 3 (1) Two dimensional Weyl fractional calculus associated with the Whittaker transform (2) Kober fractional g -integral operator of the basic analogue of the H-function CHAPTER4 (1)Some multiplication formulae for generalized hypergeometric functions (2)Study of some transformation formulas involving I-function CHAPTER 5 (1) Study of unified theorems involving the Laplace transform with application (2) Solution of a convolution integral equation with kernel as ageneralized hypergeometric function and H-function of two variables CHAPTER 6 (1) Composition of generalized fractional integrals involving product of generalized hypergeometric functions. (2) Integrals and Fourier series involving generalized hypergeometric functions CHAPTER 7 Unified study of Astrophysical thermonuclear functions forBoltzmann-Gibbs statistics and Tsallis statistics and H-function BIBLIOGRAPHY ±à¼­ÊÖ¼Ç