New series involving binomial coefficients (II)

TL;DR

The paper investigates infinite series built from binomial coefficients, particularly of the form $\displaystyle \sum_{k=1}^{\infty} \frac{a k^2 + b k + c}{k(3k-1)(3k-2) m^k \binom{4k}{k}}$, and derives several closed-form evaluations, including $\frac{2\pi}{\sqrt{3}}$, $\frac{3\pi}{2}$, and $-3\log 2$, among others. The main technique employs beta-function representations and Euler gamma integrals to transform and telescope finite-sum identities, enabling exact limits for suitable $m$ and yielding multiple identities. In addition to proving key theorems (Th1.1–Th1.3) that supply several concrete closed forms, the paper surveys and poses numerous conjectural binomial-sum identities, often involving harmonic numbers and p-adic congruences, broadening the landscape of Ramanujan-type series and their connections to constants like $\pi$, $\log$, and Catalan-like values. The work thus advances both rigorous closed-form results and a rich set of conjectures, guiding future explorations in hypergeometric transformations and related number-theoretic structures.

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 $$\sum_{k=1}^\infty\frac{(5k^2-4k+1)8^{k}}{k(3k-1)(3k-2)\binom{4k}k}=\frac{3}2π$$ 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 $$\sum_{k=0}^\infty\frac{\binom{2k}k^3}{4096^k}\left(9(42k+5)\sum_{0\le j<k}\frac1{(2j+1)^4}+\frac{25}{(2k+1)^3}\right)=\frac 56π^3.$$

Theorems & Definitions (93)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1