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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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    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?2012-08-31
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    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.2012-09-01
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    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.2012-09-01
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    Well, the motivation is hard to explain. It came up as an intermediate calculation. If its ok, can you help move it to math.SE? Otherwise I can merely repost it.2012-09-01
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    I cannot move your question because I am not a moderator.2012-09-01
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    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.2012-09-01
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    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.2012-09-01
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    I fed this into Mathematica and got: $\binom{n}{r-t} \, _3F_2(1,1,-n+r-t;r+1,r-t+1;-1)$ Note that $k$ has disappeared.2012-09-01
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    @FredDanielKline I am a bit new to this area - could you tell me what the function $_3F_2$ is called so that I may look it up?2012-09-01
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    @Ankur, it is called **HypergeometricPFQ** and per J.M. it is the preferred form. (I'm new in this area, too.)2012-09-01
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    @J.M., could you comment on my comments?2012-09-01
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    @Fred, Well, I wouldn't call it "preferred", *per se*, but I'd say there are times the series is easier to manipulate, and there are times where the hypergeometric function is easier to manipulate. OP's situation just seemed to be in the second category...2012-09-01
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    Thanks J.M and Fred. I know very little about this kind of math. I'd appreciate if someone can help me derive an expression for this sum. That way I will also be educated on the techniques used.2012-09-01
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    @Ankur Do you mean nonpositive as in $t\le 0$? So we assume $k\ge r$?2016-11-14
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    @ThomasAhle Yes.2016-12-06

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