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      Global Pad\'e approximations of the generalized Mittag-Leffler function and its inverse

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          Abstract

          This paper proposes a global Pad\'{e} approximation of the generalized Mittag-Leffler function \(E_{\alpha,\beta}(-x)\) with \(x\in[0,+\infty)\). This uniform approximation can account for both the Taylor series for small arguments and asymptotic series for large arguments. Based on the complete monotonicity of the function \(E_{\alpha,\beta}(-x)\), we work out the global Pad\'{e} approximation [1/2] for the particular cases \(\{0<\alpha<1, \beta>\alpha\}\), \(\{0<\alpha=\beta<1\}\), and \(\{\alpha=1, \beta>1\}\), respectively. Moreover, these approximations are inverted to yield a global Pad\'{e} approximation of the inverse generalized Mittag-Leffler function \(-L_{\alpha,\beta}(x)\) with \(x\in(0,1/\Gamma(\beta)]\). We also provide several examples with selected values \(\alpha\) and \(\beta\) to compute the relative error from the approximations. Finally, we point out the possible applications using our established approximations in the ordinary and partial time-fractional differential equations in the sense of Riemann-Liouville.

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          Author and article information

          Journal
          13 October 2013
          2015-12-04
          1310.5592 10.1515/fca-2015-0086

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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          26A33, 33E12, 35S10, 45K05
          Fractional Calculus and Applied Analysis, December 2015, Volume 18, Issue 6, pp 1492-1506
          15 pages, 4 figures, 1 table
          math.CA

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