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A semigroup-like property for discrete Mittag-Leffler functions. (English) Zbl 1292.39001
Summary: The discrete Mittag-Leffler function $$E_{\overline\alpha}(\lambda,z)$$ of order $$0<\alpha\leq 1$$, $$E_{\overline 1}(\lambda,z)=(1-\lambda)^{-z}$$, $$\lambda\neq 1$$, satisfies the nabla Caputo fractional linear difference equation ${}^C\nabla^\alpha_0x(t)=\lambda x(t),\quad x(0)=1,\quad t\in\mathbb N_1=\{1,2,3,\dots\}.$ Computations can show that the semigroup identity $E_{\overline\alpha}(\lambda,z_1)E_{\overline\alpha}(\lambda,z_2)=E_{\overline \alpha}(\lambda,z_1+z_2)$ does not hold unless $$\lambda=0$$ or $$\alpha=1$$. In this article we develop a semigroup property for the discrete Mittag-Leffler function $$E_{\overline\alpha}(\lambda,z)$$ in the case $$\alpha\uparrow 1$$ 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 $$\alpha\in(0,1)$$.

##### MSC:
 39A06 Linear difference equations 39A12 Discrete version of topics in analysis 33E12 Mittag-Leffler functions and generalizations 34A08 Fractional ordinary differential equations
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