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On asymptotics of discrete Mittag-Leffler function. (English) Zbl 1349.33017
On the base of the backward fractional $$h$$-sum $\nabla_h^{-\mu} f(t_n) := \frac{h}{\Gamma_h(\mu)} \sum\limits_{k=1}^{n} (t_{n-k+1})_h^{(\mu-1)} f(t_k),\tag{1}$ the following fractional $$h$$-differences are considered
– the Riemann-Liouville backward fractional $$h$$-differences ${}_{\text{R-L}} \nabla_h^{\alpha} f(t_n) := \nabla_h \nabla_h^{-(1 - \alpha)} f(t_n),\tag{2}$ – the Caputo backward fractional $$h$$-differences ${}_{\text{C}} \nabla_h^{\alpha} f(t_n) := \nabla_h^{-(1 - \alpha)} \nabla_h f(t_n).\tag{3}$ The solutions of the corresponding fractional difference equations are given in terms of the discrete Mittag-Leffler function $E_{\alpha,\beta}^{\lambda,h} := \sum\limits_{k=0}^{\infty} \lambda^k \frac{(t_n)_{h}^{\alpha k + \beta - 1}}{\Gamma(\alpha k + \beta)},\tag{4}$ with $$\lambda$$ belonging to a certain set (such that for those $$\lambda$$ the series in (4) converges).
##### MSC:
 33E12 Mittag-Leffler functions and generalizations 34A08 Fractional ordinary differential equations 39A12 Discrete version of topics in analysis
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