Around the Gauss theorem on the values of Euler’s digamma function at rational points

Abstract:

The paper is devoted to new representations of generating functions for the values of the Riemann zeta function at odd points and for certain related numbers in terms of integrals of trigonometric functions depending on a parameter $a$. In particular, new integral representations for the Euler digamma function $\psi (a)$ are obtained. The resulting integrals can be calculated in terms of the hypergeometric series ${}_3F_{2}$ and ${}_4F_{3}$ for some values of the parameters and $z=1$. Moreover, if $a$ is a proper rational fraction, then the integrals in question can be reduced to integrals of $R(\sin x, \cos x)$, where $R$ is a rational function of two variables, and are calculated explicitly. In this case, various analogs of the Gauss theorem on the values of the function $\psi (a)$ at rational points (and also yet another proof of that theorem) are obtained.