Work

Revisiting Ramanujan’s Master Theorem

Year: 2026

Type: article

Abstract: 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.

Source: Symmetry

DOI: 10.3390/sym18071208

Authors: Manuel Duarte Ortigueira, Gabriel Bengochea

Institutions: University of Lisbon, Uninova, Universidade Nova de Lisboa, Universidad Autónoma de la Ciudad de México

Language: English

Topic: Fractional Differential Equations Solutions

Subfield: Modeling and Simulation

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,896

Funders: Universidad Autónoma de la Ciudad de México, Fundação para a Ciência e a Tecnologia