Evaluations of some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$

TL;DR

This paper evaluates specific infinite series involving binomial coefficients using the beta function, deriving formulas for sums, including those related to logarithms, and proposing conjectures on related series.

Contribution

It introduces new evaluation methods for series with binomial coefficients using the beta function and provides explicit formulas and conjectures for related series.

Findings

Derived explicit formulas for series involving binomial coefficients

Established a formula for computing log n for 1<n≤85/4

Proposed conjectures on series with complex binomial summands

Abstract

In this paper, via the beta function we evaluate some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$. For example, we prove that $$\sum_{k=0}^\infty\frac{(49k+1)8^k}{3^k\binom{3k}k}=81+16\sqrt3\,\pi \ \ \text{and}\ \ \sum_{k=0}^\infty\frac{10k-1}{\binom{4k}{2k}}=\frac{4\sqrt 3}{27}\pi.$$ We also establish the following efficient formula for computing $\log n$ with $1<n\le 85/4$: \begin{align*} &\sum_{k=0}^\infty\frac{(2(n^2+6n+1)^2(n^2-10n+1)k+P(n))(n-1)^{4k}} {(-n)^k(n+1)^{2k}\binom{4k}{2k}}\\ \ \ &=6n(n+1)(n-1)^3\log n-32n(n+1)^2(n^2-4n+1), \end{align*} where $$P(n):=n^6-58n^5+159n^4+52n^3+159n^2-58n+1.$$ In addition, we pose some conjectures on series whose summands involve $\binom{2k}k/(\binom{3k}k\binom{6k}{3k})\ (k\in\mathbb N)$.