New series involving binomial coefficients (II)

TL;DR

This paper evaluates complex infinite series involving binomial coefficients and polynomial terms, providing explicit formulas and conjectures, thereby advancing the understanding of series related to binomial sums and special constants.

Contribution

It derives explicit evaluations of binomial coefficient series and introduces new conjectural identities involving these series and special constants.

Findings

Explicit formulas for series involving binomial coefficients and polynomial terms.

Proofs of series equalities involving π and logarithms.

Conjectural identities linking binomial sums to π and ζ constants.

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 $$\sum_{k=0}^\infty\frac{\binom{2k}k^3}{4096^k}\left(9(42k+5)\sum_{0\le j<k}\frac1{(2j+1)^4}+\frac{25}{(2k+1)^3}\right)=\frac 56\pi^3.$$