Certain fractional derivative formulae involving the product of a general class of polynomials and the multivariableH-function

Abstract

In the present paper, we obtain three unified fractional derivative formulae (FDF). The first involves the product of a general class of polynomials and the multivariableH-function. The second involves the product of a general class of polynomials and two multivariableH-functions and has been obtained with the help of the generalized Leibniz rule for fractional derivatives. The last FDF also involves the product of a general class of polynomials and the multivariableH-function but it is obtained by the application of the first FDF twice and it involves two independent variables instead of one. The polynomials and the functions involved in all our fractional derivative formulae as well as their arguments which are of the typex ^ρ Π ^s _i=1 (x ^t _i +α _i )^σ _i are quite general in nature. These formulae, besides being of very general character have been put in a compact form avoiding the occurrence of infinite series and thus making them useful in applications. Our findings provide interesting unifications and extensions of a number of (new and known) results. For the sake of illustration, we give here exact references to the results (in essence) of five research papers [2, 3,10, 12, 13] that follow as particular cases of our findings. In the end, we record a new fractional derivative formula involving the product of the Hermite polynomials, the Laguerre polynomials and the product ofr different Whittaker functions as a simple special case of our first formula.