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There are several identities that involve the sum of the product of binomial coefficients. However what I am searching for is an identity that involves the ratio of binomial coefficients. Specifically, I want to find a closed form expression for the sum $\sum_{k=r-t}^n\frac{\binom{n}{k}}{\binom{k+t}{r}}, $ where $n,r \in \mathbb{N}$ are fixed and $t$ is nonpositive and fixed.

Are there any standard formulae/identities that give this or are there methods for finding this sum?

I have restricted $t$ to be nonpositive, because the case where $t$ is positive is comparatively easier (in particular, $r=1, t=1$ is straight-forward to evaluate).

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    @ThomasAhle Yes.2016-12-06

2 Answers 2

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There is a technique known as Gosper's algorithm and another technique which is known as Zeilberger's algorithm. The two algorithms tackle these kinds of problems, if they succeed they will give a closed form formulas for the finite sum.

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Take a look at Petkovsek and Wilf's book "A=B", they cover techniques for such hypergeometric sums in full detail. There are packages for their algorithms (which normally are much to messy for hand computation) for leading computer algebra packages.