Evaluations of three series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{4k}{2k}$

Zhi-Wei Sun

Published 2022-04-11Version 1

In this paper, via the beta function we obtain the following three new identities: $$\sum_{k=0}^\infty\frac{k4^k}{\binom{4k}{2k}}=\frac{3\pi+8}{12}, \ \ \ \ \sum_{k=0}^\infty\frac{(30k-7)(-2)^k}{\binom{4k}{2k}}=-\frac{3\pi+64}{6},$$ and $$\sum_{k=0}^\infty\frac{14k-5}{4^k\binom{4k}{2k}}=\frac{16}{81}(\log2-24).$$