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