Two discrete Mittag-Leffler extensions of the Cayley-exponential function

Abstract

Nabla discrete fractional Mittag-Leffler (ML) functions are the key of discrete fractional calculus within nabla analysis since they extend nabla discrete exponential functions. In this article, we define two new nabla discrete ML functions depending on the Cayley-exponential function on time scales. While, the nabla discrete ML function $ E_{\overline{\gamma}} (\lambda, t) $ converges for $ |\lambda| < 1 $, both of the defined discrete functions converge for more relaxed $ \lambda $. The nabla discrete Laplace transforms of the newly defined functions are calculated and confirmed as well. Some illustrative graphs for the two extensions are provided.