Infinite and finite series involving central binomial coefficients and closed forms of generalized hypergeometric functions

Abstract

Let $\mathbb{Z}^-=\setminus\{-1,-2,\dotsc\}$. In 2023, Qi and Lim gave two claims for summing the infinite series $$ \sum_{k=1}^{\infty} \binom{2k}{k} \frac{1}{\alpha+k} \biggl(\frac{\pm1}{4}\biggr)^k, \quad \alpha\in\mathbb{C}\setminus\mathbb{Z}^-. $$ In present paper, the authors establish several sum functions of the infinite and finite series $$ \sum_{k=1}^{\infty}\binom{2k}{k}\frac{1}{\alpha+k}\biggl(\frac{z}{4}\biggr)^k \quad\text{and}\quad \sum_{k=1}^{n}\binom{2k}{k}\frac{1}{\alpha+k}\biggl(\frac{z}{4}\biggr)^k $$ for $\alpha\in\mathbb{C}\setminus\mathbb{Z}^-$ and $n\in\mathbb{N}=\{1,2,\dotsc\}$ in terms of the Gauss hypergeometric functions ${}_2F_1$ and the generalized hypergeometric functions ${}_3F_2$ for $\alpha\in\mathbb{C}\setminus\mathbb{Z}^-$ and $n\in\mathbb{N}$. In light of the Euler integral representation of the Gauss hypergeometric function ${}_2F_1$, the author present several closed forms of two Gauss hypergeometric functions ${}_2F_1$, two generalized hypergeometric functions ${}_3F_2$, and the classical incomplete beta functions $B_z\bigl(\frac12, \frac{1}{2}+n\bigr)$ and $B_z\bigl(\frac12, 1+n\bigr)$. With the help of the Euler hypergeometric transform, the authors derive closed forms of five Gauss hypergeometric functions. In addition, the authors also obtain a closed form of the differential operator $\bigl[(1-z)\frac{\operatorname{d}}{\operatorname{d}z}(1-z)\bigr]^n \frac{\arcsin\sqrt{z}}{\sqrt{z(1-z)}}$ for $n\in\mathbb{N}_0=\{0\}\cup\mathbb{N}$.