Recurrence Relations for Values of the Riemann Zeta Function in Odd Integers

In the first section we show that $\frac{\cosh(x)}{\sinh(x)^{2N+1}}$ can be expressed as linear combination of $\frac{\sinh(x)}{\cosh(x)^{2N+1}}$ and $\frac{\cosh(2x)}{\sinh(2x)^{2k+1}}$ for some $k\leq N$. We achieve this by proving four identities between certain rational functions. Then we show that the $2n$-th derivative of $\coth(x)$ can be expressed as linear combination of $\frac{\cosh(x)}{\sinh(x)^{2k+1}}$ with $k$ ranging from $1$ to $n$. We prove some useful recurrence relations between the coefficients of the $\frac{\cosh(x)}{\sinh(x)^{2k+1}}$’s and compute explicitly the inverse of the matrix formed by these coefficients. We derive our main result Theorem 3.3 - the limit identity for $\lim_{\alpha\to 0_{+}}\sum_{n=1}^{\infty}\frac{1}{n}\frac{\sinh(\alpha n)}{\cosh(\alpha n)^{2N+1}}$ for a fixed $N\in\mathbb{N}$ - by applying our previous findings on Ramanujan’s famous identity for the Riemann zeta function values in odd integers. As an application we finally determine recurrence relations of the form $\sum_{k=1}^{\infty}r_{k}\frac{\zeta(2k+1)}{\pi^{2k}}=0$.