A carregar...
Publicação
Revisiting Ramanujan’s Master Theorem
| Nome: | Descrição: | Tamanho: | Formato: | |
|---|---|---|---|---|
| 311.11 KB | Adobe PDF |
Orientador(es)
Resumo(s)
Ramanujan’s master theorem is considered to express particular values of the Liouville fractional derivatives. This fact is used to immediately obtain generalizations. Firstly, it is shown that the original formulation is suitable only for anti-causal functions. Consequently, an expression for causal functions is introduced. Secondly, the two formulations are extended to be valid for any real order and related to the Laplace transform, corresponding to transforms with left- or right-sided regions of convergence. This leads to introducing the concept of the unilateral MacLaurin series and the substitution of the Liouville derivative by the (Liouville–)Grünwald–Letnikov derivative, which essentially expresses a discrete formulation that is suitable for numerical implementations. The way in which this procedure arose and its relationship with the Liouville derivatives suggested other approaches, which are briefly discussed, such as replacing the MacLaurin series with the Mittag–Leffler series or the Liouville derivative with the Riesz, Feller, or Hadamard derivatives.
Descrição
The first author was funded by Portuguese National Funds through the FCT—Foundation for Science and Technology—within the scope of the CTS Research Unit—Center of Technology and Systems/UNINOVA/FCT/NOVA—under reference UIDB/00066/2025. The second author was supported by the Autonomous University of Mexico City (UACM) under reference UACM CCYT2026-CON-01. Publisher Copyright: © 2026 by the authors.
Palavras-chave
Grünwald–Letnikov derivative Liouville derivative Ramanujan’s master theorem Computer Science (miscellaneous) Chemistry (miscellaneous) General Mathematics Physics and Astronomy (miscellaneous)