Convolution-to-sum identities for Mittag-Leffler type functions
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arXiv
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| Format: | Preprint |
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2026
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| author | Cvetko, William Cherkaev, Elena |
| author_facet | Cvetko, William Cherkaev, Elena |
| contents | 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, $R_{α,v}$ and $P_{α,w}$. The convolution of two such functions can be expressed as a series of them. Moreover, if the functions share the same order $α$, 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 $α_1,α_2$ are related by a rational factor. For $\frac{α_1}{α_2} = \frac{n}{m}$, 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01079 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convolution-to-sum identities for Mittag-Leffler type functions Cvetko, William Cherkaev, Elena Analysis of PDEs Mathematical Physics 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, $R_{α,v}$ and $P_{α,w}$. The convolution of two such functions can be expressed as a series of them. Moreover, if the functions share the same order $α$, 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 $α_1,α_2$ are related by a rational factor. For $\frac{α_1}{α_2} = \frac{n}{m}$, 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). |
| title | Convolution-to-sum identities for Mittag-Leffler type functions |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2605.01079 |