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

Abstract

Let ${\mathbb{Z}}^{-}=\setminus \{-1,-2,\dots \}$. In 2023, Qi and Lim gave two claims for summing the infinite series $$\sum _{k=1}^{\infty }\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\frac{1}{\alpha +k}(\frac{\pm 1}{4}{)}^k,\alpha \in \mathbb{C}\setminus {\mathbb{Z}}^{-}.$$ In present paper, the authors establish several sum functions of the infinite and finite series $$\sum _{k=1}^{\infty }\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\frac{1}{\alpha +k}\left(\frac{z}{4}{)}^k\text{and}\sum _{k=1}^n\right(\genfrac{}{}{0ex}{}{2k}{k}\left)\frac{1}{\alpha +k}\right(\frac{z}{4}{)}^k$$ for $\alpha \in \mathbb{C}\setminus {\mathbb{Z}}^{-}$ and $n\in \mathbb{N}=\{1,2,\dots \}$ in terms of the Gauss hypergeometric functions ${}_2{F}_1$ and the generalized hypergeometric functions ${}_3{F}_2$ 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 ${}_2{F}_1$, the author present several closed forms of two Gauss hypergeometric functions ${}_2{F}_1$, two generalized hypergeometric functions ${}_3{F}_2$, and the classical incomplete beta functions $B_z(\frac{1}{2},\frac{1}{2}+n)$ and $B_z(\frac{1}{2},1+n)$. 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 $[(1-z)\frac{d}{dz}(1-z){]}^n\frac{\arcsin \sqrt{z}}{\sqrt{z(1-z)}}$ for $n\in {\mathbb{N}}_0=\{0\}\cup \mathbb{N}$.

20 pages