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.
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2 Answers
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Treating both sides as coefficients of $x^k$ in a power series with variable $x$, the left-hand side turns into a power-series product, one factor representing $\sqrt{1-x}$. We end up with a special case of the known hypergeometric identity $$(1-x)^{a+b-c} {}_2F_1(a,b;c;x) = {}_2F_1(c-a,c-b;c;x)$$ with $a = m+1$, $b = -m$, $c = 1/2$. Cf. (34) of this MathWorld entry.
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Equivalently to ccorn's answer, the sum may be written as $$\frac {(-1/2)_{k}}{k!}{}_{3}F_{2}\left({{m+1,-m,-k}\atop {3/2-k,1/2}} \Bigm |1\right)$$ which may be evaluated by Saalschütz's theorem (also called the Pfaff-Saalschütz theorem).
