Finite Sum Representation of Partial Derivatives of Multivariable Incomplete Aleph Functions

Abstract

Newly discovered, incomplete forms of special functions are increasing the interest of both pure and applied mathematicians. The main purpose of this work is to derive four theorems on partial derivatives with incomplete Aleph functions of two variables and generalize them up to r-variables. In addition to these theorems, we also established some novel formulae on the partial derivatives that play a key role in deriving the main results in terms of finite sum. Further, we generalize the result and obtain the finite sum for the incomplete Aleph functions with r-variables. Here, we also established some particular cases that are in most general character and including the results given earlier by Buschman and Deshpande and may prove significant in numerous interesting situations appearing in the literature on mathematical analysis, applied mathematics and mathematical physics.