Skip to main content
MathOverflow

Return to Question

deleted 1 characters in body
Source Link
TCL
TCL
  • 744
  • 5
  • 14

Let $m,k$ be positive integers with $k\le m$. Does anyone know some hypergeometric identities that imply $$\sum_{j=0}^k\frac{(-1/2)_{k-j}(m+1)_j(-m)_j}{(1/2)_j(k-j)!j!} =\frac{(-m-\frac{1}{2})_k(m+\frac{1}{2})_k}{(\frac{1}{2})_kk!}$$ where $(a_k=a(a+1)\cdots (a+k-1)$$a_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer's symbol.

Let $m,k$ be positive integers with $k\le m$. Does anyone know some hypergeometric identities that imply $$\sum_{j=0}^k\frac{(-1/2)_{k-j}(m+1)_j(-m)_j}{(1/2)_j(k-j)!j!} =\frac{(-m-\frac{1}{2})_k(m+\frac{1}{2})_k}{(\frac{1}{2})_kk!}$$ where $(a_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer's symbol.

Let $m,k$ be positive integers with $k\le m$. Does anyone know some hypergeometric identities that imply $$\sum_{j=0}^k\frac{(-1/2)_{k-j}(m+1)_j(-m)_j}{(1/2)_j(k-j)!j!} =\frac{(-m-\frac{1}{2})_k(m+\frac{1}{2})_k}{(\frac{1}{2})_kk!}$$ where $a_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer's symbol.

Source Link
TCL
TCL
  • 744
  • 5
  • 14

Hypergeometric identities

Let $m,k$ be positive integers with $k\le m$. Does anyone know some hypergeometric identities that imply $$\sum_{j=0}^k\frac{(-1/2)_{k-j}(m+1)_j(-m)_j}{(1/2)_j(k-j)!j!} =\frac{(-m-\frac{1}{2})_k(m+\frac{1}{2})_k}{(\frac{1}{2})_kk!}$$ where $(a_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer's symbol.