涉及中心二项式系数的无穷级数与有限级数及广义超几何函数的闭式
Infinite and finite series involving central binomial coefficients and closed forms of generalized hypergeometric functions
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中文总结 AI 辅助
本文针对含中心二项式系数的无穷与有限级数，利用高斯超几何函数的欧拉积分表示、欧拉超几何变换，推导了相关超几何函数及不完全贝塔函数的闭式，还得到了特定微分算子的闭式。
中文摘要 AI 辅助
设$\boldsymbol{\text{Z}}^- = \boldsymbol{\text{Z}} \backslash \boldsymbol{\text{Z}}^+$（即负整数集）。2023年，Qi和Lim给出了无穷级数$\boldsymbol{\text{sum}}_{k=1}^{\boldsymbol{\text{infty}}} \binom{2k}{k} \frac{1}{\boldsymbol{\text{alpha}}+k} \biggl(\frac{\boldsymbol{\text{pm}}1}{4}\biggr)^k$（其中$\boldsymbol{\text{alpha}} \boldsymbol{\text{in}} \boldsymbol{\text{C}} \backslash \boldsymbol{\text{Z}}^-$）求和的两个结论。本文作者针对$\boldsymbol{\text{alpha}} \boldsymbol{\text{in}} \boldsymbol{\text{C}} \backslash \boldsymbol{\text{Z}}^-$、$\boldsymbol{n} \boldsymbol{\text{in}} \boldsymbol{\text{N}} = \boldsymbol{\text{\text{1,2,}}}\boldsymbol{\text{dotsc}}$的情况，建立了无穷级数$\boldsymbol{\text{sum}}_{k=1}^{\boldsymbol{\text{infty}}}\binom{2k}{k}\frac{1}{\boldsymbol{\text{alpha}}+k}\biggl(\frac{\boldsymbol{z}}{4}\biggr)^k$与有限级数$\boldsymbol{\text{sum}}_{k=1}^{\boldsymbol{n}}\binom{2k}{k}\frac{1}{\boldsymbol{\text{alpha}}+k}\biggl(\frac{\boldsymbol{z}}{4}\biggr)^k$的若干求和函数，结果用高斯超几何函数${}_2F_1$和广义超几何函数${}_3F_2$表示。基于高斯超几何函数${}_2F_1$的欧拉积分表示，作者给出了两个高斯超几何函数${}_2F_1$、两个广义超几何函数${}_3F_2$以及经典不完全贝塔函数$B_z\bigl(\frac12, \frac{1}{2}+\boldsymbol{n}\bigr)$和$B_z\bigl(\frac12, 1+\boldsymbol{n}\bigr)$的若干闭式；借助欧拉超几何变换，推导了五个高斯超几何函数的闭式；此外，还得到了微分算子$\bigl[(1-\boldsymbol{z})\frac{\boldsymbol{\text{d}}}{\boldsymbol{\text{d}}\boldsymbol{z}}(1-\boldsymbol{z})\bigr]^\boldsymbol{n} \frac{\boldsymbol{\text{arcsin}}\boldsymbol{\text{sqrt}}\boldsymbol{z}}{\boldsymbol{\text{sqrt}}\boldsymbol{z(1-\boldsymbol{z})}}$（其中$\boldsymbol{n} \boldsymbol{\text{in}} \boldsymbol{\text{N}}_0 = \boldsymbol{\text{\text{0}}}\boldsymbol{\text{cup}}\boldsymbol{\text{N}}$）的一个闭式。
英文摘要
Let $\mathbb{Z}^-=\setminus\{-1,-2,\dotsc\}$. In 2023, Qi and Lim gave two claims for summing the infinite series $$ \sum_{k=1}^{\infty} \binom{2k}{k} \frac{1}{α+k} \biggl(\frac{\pm1}{4}\biggr)^k, \quad α\in\mathbb{C}\setminus\mathbb{Z}^-. $$ In present paper, the authors establish several sum functions of the infinite and finite series $$ \sum_{k=1}^{\infty}\binom{2k}{k}\frac{1}{α+k}\biggl(\frac{z}{4}\biggr)^k \quad\text{and}\quad \sum_{k=1}^{n}\binom{2k}{k}\frac{1}{α+k}\biggl(\frac{z}{4}\biggr)^k $$ for $α\in\mathbb{C}\setminus\mathbb{Z}^-$ 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 $α\in\mathbb{C}\setminus\mathbb{Z}^-$ 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 $n\in\mathbb{N}_0=\{0\}\cup\mathbb{N}$.
