Some Operational Formulas Involving the Operators $xD$, $x\Delta $ and Fractional Derivatives

Abstract

We consider the operator $\{ D^{1 - \delta } \prod _{j = 1}^m (xD + \alpha _j )^r x^\delta \} ^n $ ($\delta = 0 {\text{ or }}1$; m, n, r integers, and $\alpha _j $ arbitrary) which contains as a special case the operator $(D(xD)^r )^n $ previously studied by Carlitz. We also consider the analogous operator involving the finite difference operator $\Delta $. Some general operational formulas are established from which interesting relationships may be deduced. Further generalizations of operational formulas for the fractional derivative are also given. In particular a law of exponents is deduced for the general fractional operator $D_z^\alpha (z^\alpha D_z^\alpha )^r $ where $\alpha $ may be nonintegral.