Evaluations of some series of the type $\sum_{k=0}^\infty(ak+b)x^k/\binom{mk}{nk}$
Zhi-Wei Sun
TL;DR
This work develops a beta-function framework to evaluate series of the form $S_{m,n}(a,b,x)=\sum_{k=1}^\infty (a k + b)\,x^k/\binom{m k}{n k}$ for integers $m>n>0$ and suitable $x$, establishing convergence criteria and a Beta/Gamma-based integral representation. It then specializes to the important cases $(m,n)=(3,1)$ and $(m,n)=(4,2)$, obtaining closed-form evaluations in terms of $\arctan$, $\arccot$, $\arctanh$-type expressions and the auxiliary function $R(x)$, which enable explicit Ramanujan-type sums such as $\sum_{k\ge0}\frac{(49k+1)8^k}{3^k\binom{3k}{k}}=81+16\sqrt{3}\,\pi$ and $\sum_{k\ge0}\frac{10k-1}{\binom{4k}{2k}}=\frac{4\sqrt{3}}{27}\,\pi$. A central result is a beta-integral representation for these sums, from which a rapid $\log n$ expansion is derived and valid for $1<n\le 85/4$, with a polynomial $P(n)$ guiding the logarithmic term. The paper also presents a conjectural program exploring series with summands involving $\binom{2k}{k}/(\binom{3k}{k}\binom{6k}{3k})$ and related arithmetic properties, including $p$-adic congruences and connections to Euler numbers and zeta values. Overall, the work broadens the toolbox for evaluating binomial-sum series and producing rapidly convergent series for $\pi$ and logarithms, with potential applications in high-precision computations and number theory.
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 $$\sum_{k=0}^\infty\frac{(49k+1)8^k}{3^k\binom{3k}k}=81+16\sqrt3\,π\ \ \text{and}\ \ \sum_{k=0}^\infty\frac{10k-1}{\binom{4k}{2k}}=\frac{4\sqrt 3}{27}π.$$ 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 $\binom{2k}k/(\binom{3k}k\binom{6k}{3k})\ (k\in\mathbb N)$.