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Context: This post is related to Conjectured closed forms for Eisenstein-like series . This is an extension of it. We have also: \begin{align*} S(x):=\sum_{n=1}^{\infty}\frac{(2n-1)\left(e^{\frac{\pi(2n-1)}{2x}}\left(\cos{(\frac{\pi}{2x})}\cos{(\frac{\pi n}{x})}+\sin{(\frac{\pi}{2x})}\sin{(\frac{\pi n}{x})}\right)+1 \right)}{e^{\frac{\pi(2n-1)}{x}}+2e^{\frac{\pi(2n-1)}{2x}}\left(\cos{(\frac{\pi}{2x})}\cos{(\frac{\pi n}{x})}+\sin{(\frac{\pi}{2x})}\sin{(\frac{\pi n}{x})} \right)+1}=\frac{1}{24},\tag{1} \end{align*} for all $x \in \mathbb{N}$. For $x=1$, the sum reduces to: \begin{align*} \sum_{n=1}^{\infty}\frac{(2n-1)}{e^{\pi(2n-1)}+1}=\frac{1}{24}, \tag{2} \end{align*} which is well known and in the linked question we can find a proof. Also several different proofs here Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

For $x=2$, the sum is: \begin{align*} \sum_{n=1}^{\infty}\frac{(2n-1)\left( e^{\frac{\pi(2n-1)}{4}}\left( \cos{(\frac{\pi n}{2})+\sin{(\frac{\pi n}{2})}} \right) +\sqrt{2}\right)}{\sqrt{2}e^{\frac{\pi(2n-1)}{2}}+2e^{\frac{\pi(2n-1)}{4}}\left(\cos{(\frac{\pi n}{2})+\sin{(\frac{\pi n}{2})}} \right)+\sqrt{2}}=\frac{1}{24}. \tag{3} \end{align*} Deal with $(1)$ seems to be possible as Paramanand Singh dealt with the previous and this is so close. I have a systematic way to obtain sums for which $S(x)=S(x+1)$ when $x$ is a positive integer.

Question: Is there an elementary way (avoiding modular forms) for proving $S(x)=S(x+1)$ when $x$ is a positive integer for functions like $S(x)$?

User
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    The above problem is similar to the one in your previous post. Use the values $\tau=i/4x,u=(i+1)/4x,v=(i-1)/4x$ and $q=\exp(2\pi i\tau) $ and the sum in question is $\frac{q} {2}\frac{d}{dq}\log\prod_{n\geq 1}(1+\exp((2n-1)2\pi iu))(1+\exp((2n-1)2\pi iv))$ and one can then express it in terms of $E_2$. – Paramanand Singh Aug 21 '23 at 03:03
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    The key part here in such evaluations is that $(i+1)(i-1)=-2$ and more importantly $(i+1)^2+(i-1)^2=0$ which permits closed form without evaluation of $E_2(ix)$. – Paramanand Singh Aug 21 '23 at 03:47
  • Perhaps you should also discuss your approach to generate such sums which are constant at integers. – Paramanand Singh Aug 21 '23 at 05:19
  • Hi @ParamanandSingh as soon as possible I will improve the question with the way I obtain sums that are constant at positive integers. – User Aug 21 '23 at 05:53

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