The unified Riemann-Liouville fractional derivative formulae

Abstract

In this paper, we obtain two unified fractional derivative formulae. The first involves the product of two general class of polynomials and the multivariable $H$-function. The second fractional derivative formula also involves the product of two general class of polynomials and the multivariable $H$-function and has been obtained by the application of the first fractional derivative formula twice and it has two independent variables instead of one. The polynomials and the functions involved in both the fractional derivative formulae as well as their arguments are quite general in nature and so our findings provide interesting unifications and extensions of a number of (known and new) results. For the sake of illustration, we point out that the fractional derivative formulae recently obtained by Srivastava, Chandel and Vishwakarma [11], Srivastava and Goyal [12], Gupta, Agrawal and Soni [4], Gupta and Agrawal [3] follow as particular cases of our findings. In the end, we record a new fractional derivative formula involving the product of the Konhauser biorthogonal polynomials, the Jacobi polynomials and the product of $r$ different modified Bessel functions of the second kind as a simple special case of our first formula.