New series involving binomial coefficients (II)

Abstract

In this paper, we evaluate some series of the form $$\sum_{k=1}^\infty\frac{ak^2+bk+c}{k(3k-1)(3k-2)m^k\binom{4k}k}.$$ For example, we prove that $$\sum_{k=1}^\infty\frac{(5k^2-4k+1)8^{k}}{k(3k-1)(3k-2)\binom{4k}k}=\frac{3}2\pi$$ and $$\sum_{k=1}^\infty\frac{415k^2-343k+62}{k(3k-1)(3k-2)(-8)^k\binom{4k}k}=-3\log2.$$ We also pose many new conjectural series identities involving binomial coefficients; for example, we conjecture that $$\sum_{k=0}^\infty\frac{\binom{2k}k^3}{4096^k}\left(9(42k+5)\sum_{0\le j<k}\frac1{(2j+1)^4}+\frac{25}{(2k+1)^3}\right)=\frac 56\pi^3.$$