An Improper Multiple Integral

[benorin]

TL;DR Summary

I was writing a paper and derived some results using a theorem that turned out to be false, however some of these results are known to be true, so I am attempting to see if any of these are true, the one I mention here involves the Lerch Transcendent

The Lerch Transcendent identity from my paper which may or may not be true, for ##N\in\mathbb{Z}^+##, and I forget the domain of z and y, here it goes

$$\Phi (z,N,y) :=\sum_{q=0}^{\infty}\frac{z^q}{(q+y)^N}$$
$$=\int_{0}^{1}\int_{0}^{1}\cdots \int_{0}^{1}\prod_{k=1}^{N}\left( \lambda_k^{y-1}\right)\left( 1-z\prod_{q=1}^{N}\lambda_q\right)^{-1}\, d\lambda_1d\lambda_2\cdots d\lambda_N$$

Some of the results I got using that untrue theorem were known results that were actually true, but I didn't check all of them, I wish to try to verify this identity by other means but I'll be honest I've not done much math for about 20 years so I need a little guidance here please?

How to handle the infinite discontinuity of the integrand at ##z=\lambda_k=1## for ##k=1,2,\ldots, N##? Do I take the upper bound of each integral to be ##1-\epsilon## and let ##\epsilon\rightarrow 0+##? Or do I have to set each upper bound to be ##1-\epsilon_k## and take a N-dimensional limit? Unsure how to start.

 

[benorin]

For ##N\in\mathbb{Z}^+, |z|<1, y\in\mathbb{R}##, define

$$J=\int_{0}^{1}\int_{0}^{1}\cdots \int_{0}^{1}\prod_{k=1}^{N}\left( \lambda_k^{y-1}\right)\left( 1-z\prod_{q=1}^{N}\lambda_q\right)^{-1}\, d\lambda_1d\lambda_2\cdots d\lambda_N$$
$$=\lim_{(\epsilon_1,\epsilon_2,\ldots,\epsilon_N )\rightarrow (0,0,\ldots, 0)}\int_{0}^{1-\epsilon_N}\int_{0}^{1-\epsilon_{N-1}}\cdots \int_{0}^{1-\epsilon_1}\sum_{q=0}^{\infty}z^q\prod_{k=1}^{N}\left( \lambda_k^{y+q-1}\right) \, d\lambda_1d\lambda_2\cdots d\lambda_N$$
$$=\lim_{(\epsilon_1,\epsilon_2,\ldots,\epsilon_N )\rightarrow (0,0,\ldots, 0)}\sum_{q=0}^{\infty}\frac{z^q}{(y+q)^N}\prod_{k=1}^{N}\left( 1-\epsilon_k\right) ^{y+q} =\sum_{q=0}^{\infty}\frac{z^q}{(q+y)^N} =:\Phi (z,N,y)$$

Conceivably one may take ##z,y\in\mathbb{C}## such that ##|z|<1##? My complex analysis is rusty to say the least, I can look it up later I guess. So my identity holds, just got to hammer out the domains. I used the N-dimensional limit as being the more general possibility, to cover all my bases. Was there any flaw in my work?

 

[benorin]

*bump*