New series involving binomial coefficients

Zhi-Wei Sun

Published 2023-07-06Version 1

In this paper, we deduce several new identities on infinite series with denominators of summands containing both binomial coefficients and linear parts. We mainly evaluate the sums $$\sum_{k=1}^\infty\frac{x_0^k}{(2k-1)\binom{3k}k},\ \sum_{k=0}^\infty\frac{x_0^k}{(3k+1)\binom{3k}k} \ \text{and}\ \sum_{k=0}^\infty\frac{x_0^k}{(3k+2)\binom{3k}k}$$ for any $x_0\in(-27/4,27/4)$. In particular, we have $$\sum_{k=1}^\infty\frac{(\frac{9-3\sqrt3}2)^k}{(2k-1)\binom{3k}k}=(\sqrt3-1)\log(\sqrt3+1).$$ We also prove that $$\sum_{k=1}^\infty \frac{\sum_{k<j\le 3k}1/j}{(2k-1)2^k\binom{3k}k}=\frac{\pi^2}{36}-\frac{\pi}{18} -\frac2{15}(G+\log^22)+\frac{\log2}3,$$ where $G$ is the Catalan constant. In addition, we pose some new conjectural series identities involving binomial coefficients; for example, we conjecture that $$\sum_{k=1}^\infty\frac{\binom{2k}k^3}{4096^k}\bigg(9(42k+5)\sum_{j=0}^{k-1}\frac1{(2j+1)^4}+\frac{25}{(2k+1)^3}\bigg)=\frac 56\pi^3.$$