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Certain Fractional Q-Derivative Integral Formulae for the Basic Aleph-Function of Two Variables

In the present paper, we derive two theorem involving fractional q-integral operators of Erdélyi-Kober type and a basic of Aleph-function of two variables. Corresponding assertions for the Riemann-Liouville and Weyl fractional q-integral transforms are also presented. Several special cases of the main results have been illustrated in the concluding section. 1.Introduction. The fractional q-calculus is the q-extension of the ordinary fractional calculus. The subject deals with the investigations of q-integral and q-derivatives of arbitrary order, and has gained importance due to its various applications in the areas like ordinary fractional calculus, solutions of the q-difference (differential) and q-integral equations, q-transform analysis see ([1] and [10]). Motivated by these avenues of applications, a number of workers have made use of these operators to evaluate fractional q-calculus formulae for various special function, basic analogue of Fox’s H-function, general class of q-polynomials etc. One may refer to the recent paper [6]-[7], [13] and [16]-[18] on the subject. In this paper, we have established two theorems involving the fractional q-integral operator of Erdélyi-Kober type, which generalizes the Riemann-Liouville and Weyl fractional q-integral operators. Several special cases of the main results have been illustrated in the concluding section. In the theory of q-series, for real or complex and , the -shifted factorial is defined as :