what is the long time approximation from the following the asymptotic series function? $$t^{3\alpha-2}\sum_{n=0}^\infty\sum_{m=0}^\infty\frac{(-it^{\alpha})^{n}}{Γ(n\alpha+2\alpha)}\frac{(it^{\alpha})^{m}}{Γ(m\alpha+\alpha)}$$ Thanks for your advice.
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Your series is a product of Mittag-Leffler functions whose asymptotics can be found in this paper. – Antonio Vargas May 04 '18 at 01:44
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This $\texttt{arXiv paper can be useful}$. – Felix Marin May 04 '18 at 03:03
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that's mean: long time approximation for Mittag-Leffler function is $$E_{\alpha,2\alpha}(-it^{\alpha})\sim-\frac{(-it^{\alpha})^{-1}}{Γ(2\alpha-\alpha)}$$ and$$ E_{\alpha,\alpha}(it^{\alpha})\sim-\frac{(it^{\alpha})^{-1}}{Γ(\alpha-\alpha)}$$ – karen2 May 04 '18 at 08:03
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but $$E_{\alpha,\alpha}(-it^{\alpha})\sim-\frac{(it^{\alpha})^{-1}}{Γ(\alpha-\alpha)}$$ in denominator will be infinity and total phrase will be zero. can you help me in to solve this problem? thanks for your advice. – karen2 May 04 '18 at 08:12