A Further Extension of the Leibniz Rule to Fractional Derivatives and Its Relation to Parseval’s Formula

The familiar Leibniz rule for the Nth derivative of the product of two functions is DN uv = ∑ ( array*20c N \\ n \\ array )DN - n uDⁿ v. A generalization of this formula for fractional derivatives is given as D^α uv = ∑ a ( array*20c α \\ an + γ \\ array )Dα - an - γ uDan + γ v, where α need not be a natural number and 0 < a 1. (The special case, $a = 1$, appeared previously.) Further generalizations of the Leibniz rule are also given and are derived from a generalization of Taylor’s series given previously by the author. It is shown that these new series are generalizations of Parseval's formula from the study of Fourier series. Finally, new series expansions relating the special functions of mathematical physics are derived as special cases of the generalizations of the Leibniz rule. These series include a generalized Dougall's formula, several series of the Cardinal type, and a series related to a problem of Ramanujan.