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Recently Chaurasia and Gill [7], Chaurasia and Kumar [8] have solved the one-dimensional integral equation of Fredholm type involving the product of special functions. We solve an integral equation involving the product of a class of multivariable polynomials, the multivariable H-function defined by Srivastava and Panda [29, 30] and the multivariable I-function defined by Prasad [21] by the application of fractional calculus theory. The results obtained here are general in nature and capable of yielding a large number of results (known and new) scattered in the literature.

The object of the paper is to establish an integral formula involving the generalized prolate spheroidal wave function, generalized hypergeometric function, generalized polynomials, Aleph-function of one variable and multivariable Aleph-function. This integral formula has been employed to obtain an expansion formula for multivariable Aleph-function, generalized hypergeometric function, a class of polynomial and Aleph-function in terms of generalized prolate spheroidal wave function.This expansion formula being of very general nature can be transformed to provide many new results involving various commonly used special functions occuring in applied mathematics, mathematics physics and mchanics. During the course of finding, we establish several particular cases.

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