Infinite and finite series involving central binomial coefficients and closed forms of generalized hypergeometric functions
Year: 2026
Type: preprint
Abstract: Let $\mathbb{Z}^-=\{-1,-2,\dotsc\}$ and $\mathbb{N}=\{1,2,\dotsc\}$. In 2023, Qi and Lim gave two claims for summing the infinite series $\sum_{k=1}^{\infty}\binom{2k}{k}\frac{1}{α+k}\bigl(\frac{\pm1}{4}\bigr)^k$ for $α\in\mathbb{C}\setminus\mathbb{Z}^-$. In this paper, we confirm these claims by expressing the infinite series $\sum_{k=1}^{\infty}\binom{2k}{k}\frac{1}{α+k}\bigl(\frac{z}{4}\bigr)^k$ and the finite series $\sum_{k=1}^{n}\binom{2k}{k}\frac{1}{α+k}\bigl(\frac{z}{4}\bigr)^k$ in terms of the Gauss hypergeometric function ${}_2F_1\bigl[\tfrac12,α;1+α;z\bigr]$ and a generalized hypergeometric function ${}_3F_2$, for $|z|\le1$ and $n\in\mathbb{N}$. Using the Euler integral representation, we derive closed forms, in terms of finite sums of sines and cosines of $\arcsin\sqrt{z}\,$, for ${}_2F_1\bigl[\tfrac12,\tfrac12+n;\tfrac32+n;z\bigr]$, ${}_2F_1\bigl[\tfrac12,1+n;2+n;z\bigr]$, two related ${}_3F_2$ functions, and the incomplete beta functions $B_z\bigl(\frac12+n,\frac12\bigr)$ and $B_z\bigl(1+n,\frac12\bigr)$. By the Pfaff transformation, we obtain closed forms of several further Gauss hypergeometric functions with parameters shifted by $n$. We also give a closed form of $\bigl[(1-z)\frac{\operatorname{d}}{\operatorname{d}z}(1-z)\bigr]^n\frac{\arcsin\sqrt{z}\,}{\sqrt{z(1-z)}\,}$ and a combinatorial identity.
Source: arXiv (Cornell University)
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7201868562