A semigroup-like Property for Discrete Mittag-Leffler Functions

AbstractDiscrete Mittag-Leffler function Eᾱ(λ,z) of order 0 < α ≤ 1, E1̄(λ,z)=(1-λ)-z, λ ≠ 1, satisfies the nabla Caputo fractional linear difference equation C∇0αx(t)=λx(t),x(0)=1,t∈ℕ1={1,2,3,…}. Computations can show that the semigroup identity Eᾱ(λ,z1)Eᾱ(λ,z2)=Eᾱ(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 Eᾱ(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).