MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 7 down vote accepted

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.

share|cite|improve this answer

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).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.