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Let $\Phi\left(z,s,a\right) $ be a Lerch Trascendent. $$\Phi\left(z,s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{t^{s-1}e^{-at}}{1-ze^{-t}}dt.$$

Can we upper bound the above in terms of its parameters (for positive reals $z, a$ and real $s$)?

Pls let me know if there is any doubt regarding the problem.

Thanks in advance!

Ram
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  • The sum representation of the same function is a simple expression in terms of the parameters that give a bound (more, the exact value). It won't get simpler as a closed expression. I'm pretty sure that for s=1, the function always explodes past z=1. – Nikolaj-K Mar 17 '17 at 12:40

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