Convolution-to-sum identities for Mittag-Leffler type functions

Abstract

Product-to-sum identities for trigonometric functions play a fundamental role in function theory and numerous applications. In this spirit, we present convolution-to-sum identities for Mittag-Leffler type functions. Using a Laplace domain analysis of fractional operators, we identify a family of Mittag-Leffler type functions that encapsulates the eigenfunctions of Riemann-Liouville and Caputo fractional derivatives. We work with two closely-related parameterizations of this class, $R_{\alpha,v}$ and $P_{\alpha,w}$. The convolution of two such functions can be expressed as a series of them. Moreover, if the functions share the same order $\alpha$, the convolution can be reduced to a sum of two $P/R$ functions through a partial-fraction decomposition in the Laplace domain. Furthermore, $R$ and $P$ functions satisfy a generalization of Euler's identity, which expands the scope of the previous result to convolutions of $P/R$ functions whose orders $\alpha_1,\alpha_2$ are related by a rational factor. For $\frac{\alpha_1}{\alpha_2} = \frac{n}{m}$, the resulting sum has $n+m$ terms. The foundational results and methods developed here are illustrated by their application to forced subdiffusion and to a fractionally attenuated wave equation (the Caputo-Wismer-Kelvin, or the fractional Kelvin-Voigt model).