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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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