论文标题
从Gegenbauer系列扩展获得的新身份
New identities obtained from Gegenbauer series expansion
论文作者
论文摘要
使用傅立叶杰根巴尔系列中的扩展，我们证明了几种延伸和推广已知结果的身份。特别是，在其他结果中证明了 \ begin {equation*} \ sum_ {n = 0}^\ infty \ frac {1} {4^n} \ binom {2n} {n} {n} \ frac {z-2n} {\ binom {z-1/2} {z-1/2} {n}}} {n}}} \ binom {z \ binom {z} n} n}^3 = \ frac {\ tan（πz）}π \ end {equation*} 对于所有复数数字$ z $，以使$ \ re（z）> - \ frac {1} {2} $和$ z \ notin \ frac {1} {2} {2}+\ m athbb {z} $。
Using the expansion in a Fourier-Gegenbauer series, we prove several identities that extend and generalize known results. In particular, it is proved among other results, that \begin{equation*} \sum_{n=0}^\infty\frac{1}{4^n}\binom{2n}{n}\frac{z-2n}{\binom{z-1/2}{n}}\binom{z}{n}^3 =\frac{\tan(πz)}π \end{equation*} for all complex numbers $z$ such that $\Re(z)>-\frac{1}{2}$ and $z\notin\frac{1}{2}+\mathbb{Z}$.
