A Semigroup-Like Property for Discrete Mittag-Leffler Functions

Abstract

Discrete Mittag-Leffler function of order 0 < alpha a parts per thousand currency sign 1, , lambda not equal 1, satisfies the nabla Caputo fractional linear difference equation (C)del(alpha)(0)(t) = lambda x(t), x(0) = 1, t is an element of N-1 = {1, 2, 3, ...}. Computations can show that the semigroup identity E alpha(lambda, z1)E alpha(lambda, z2) = E alpha(lambda, z1 + z2) does not hold unless lambda = 0 or alpha = 1. In this article we develop a semigroup property for the discrete Mittag-Leffler function in the case alpha a dagger 1 is just the above identity. The obtained semigroup identity will be useful to develop an operator theory for the discrete fractional Cauchy problem with order alpha a (0, 1).