A Semigroup-Like Property for Discrete Mittag-Leffler Functions

Abstract

Discrete Mittag-Leffler function E.ᾱ (λ, z) of order 0 <α ≤ 1, E.1̄(λ, z) = (1 - λ)-z, l ≠ 1, satisfies the nabla Caputo fractional linear difference equation C∇0α x(t) = λx(t), x(0) = 1, t ∈ ℕ1 = {1, 2, 3,.. .}. Computations can show that the semigroup identity E.ᾱ (λ, z1)E. ᾱ (λ, z2) = E.ᾱ (λ, z1 + z2) does not hold unless λ = 0 or α = 1. In this article we develop a semigroup property for the discrete Mittag-Leffler function E.ᾱ (λ, z) in the case α ↑ 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 α ∈ (0, 1).