On proportional hybrid operators in the discrete setting

Abstract

In this article, we introduce a new nonlocal operator (Formula presented.) defined as a linear combination of the discrete fractional Caputo operator and the fractional sum operator. A new dual operator (Formula presented.) is also introduced by replacing the discrete fractional Caputo operator with the discrete fractional Riemann–Liouville operator. It is shown that it corresponds to a natural discretization of a proportional hybrid operator defined by the Riemann–Liouville operator instead of Caputo hybrid operator. We then analyze the most important properties of these operators, such as their inverse operator and the (Formula presented.) -transform, among others. As an application, we solve difference equations equipped with these operators and obtain explicit solutions for them in terms of trivariate Mittag-Leffler sequences.