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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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