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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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  • $\begingroup$ Mods, since I haven't even received a comment over the last two days, would it be ok to re-post this question now on MathOverflow? $\endgroup$
    – Ankur
    Commented Aug 31, 2012 at 17:46
  • $\begingroup$ Moderators are not checking all comments to all questions. If you have a question to moderators, it is more reliable to flag your own question, choose “it needs ♦ moderator attention,” and explains why it needs moderator attention. $\endgroup$
    – Tsuyoshi Ito
    Commented Sep 1, 2012 at 0:42
  • $\begingroup$ At least for me, this question is fairly uninteresting because it looks like just a random formula, and I fail to see why it is reasonable to expect that it has a nice closed-form expression (and also because I am not good at this kind of math). If you state why you are interested in this sum, some people may care more. $\endgroup$
    – Tsuyoshi Ito
    Commented Sep 1, 2012 at 0:43
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    $\begingroup$ Usually, if you have a ratio of binomial coefficients as the terms of a sum, a useful first step is to switch to hypergeometric form, and then do the manipulations on the resulting hypergeometric functions. $\endgroup$
    – J. M. ain't a mathematician
    Commented Sep 1, 2012 at 7:54
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    $\begingroup$ Is there a typo at here? Why $C^{k+t}_{r}$ appeared as you defined $k=r-t$? This would let to $\sum C^{n}_{k}=2^{n}$ and I guess that is not what you wanted. $\endgroup$
    – Bombyx mori
    Commented Sep 1, 2012 at 8:05

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.

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