Evaluations of some Euler-Apéry-type series (Q2095055)

The subject of this paper is the Euler-Apéry-type series whose general terms is a product of central binomial coefficients, generalized harmonic numbers and \((n 4^n)^{-1}\). In particular, the authors use the method of contour integration to prove the result that the Euler-Apéry-type series \[ \sum_{n=1}^\infty \frac{n}{(n-1/2)^q}\cdot \frac{\binom{2n}{n}}{4^n} \in \mathbb{Q}[\pi, \log 2, \zeta(3), \zeta(5), \zeta(7), \ldots]. \] Furthermore, they give an explicit evaluation for the EulerApéry-type series \[ \sum_{n=1}^\infty \binom{mn}{n}\frac{x^n}{n^p} \] for positive integers \(m\) and \(p\), by using the method of generating function involving Fuss-Catalan numbers. Finally, they establish a recurrence relation for general Euler-Apéry-type series involving multiple harmonic star sum.