Some infinite sums arising from the Weierstrass product theorem
The Hurwitz zeta function is defined as \[ \zeta (s,a) = \sum_{n=0}^\infty \frac{1}{(n+a)^s} \qquad \big(s > 1, a \in\mathbb{R}\setminus \{0,-1,-2,\ldots\}\big). \] The special case \(s = 2\) results in the trigamma function \(\psi'\), where \(\psi\) denotes the digamma function. Using the well-known Weierstrass product representations for the trigonometric functions, the author proves a number of identities for infinite sums involving the Hurwitz zeta function and the digamma and trigamma function. Typical examples are \[ \sum_{n=1}^\infty \frac{\zeta (2,n)}{n^2} = \frac{7\pi^4}{360} \] (Identity 4) and \[ \sum_{n=1}^\infty \frac{\zeta (4,n)}{n^4} = \frac{13\pi^8}{113400} \] (Identity 7). Another example is Identity 11 giving a closed form for the series \[ s:=\sum\limits_{n=1}^\infty \frac{n^2 - 1/2}{(n^2 + 1/2)^2} \left[ \psi' \left( n - i/\sqrt{2}\right) + \psi' \left( n + i/\sqrt{2}\right)\right] \] namely \[ s= - 1 + {\frac{\sqrt{2}}{4} \pi \coth(\pi/\sqrt{2}) - \frac{3\pi^2}{4\sinh^2 (\pi/\sqrt{2})}} + \frac{\pi^4}{12\sinh (\pi/\sqrt{2})} \left( 5 + \cosh \big( \pi\sqrt{2}\big)\right) . \]
scientific article; zbMATH DE number 6305693