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Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.

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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+\f …