Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 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

My question reads: Does this formula has mathematical meaning at first place? Is it summable? $$\sum^{\infty}_{k=0}{n\choose k}{m\choose k} x^k$$

share|cite|improve this question
Your "infinite" sum is nothing else than a polynomial in $\,x\,$ of degree $\,\max(n,m)\,$ – DonAntonio Oct 11 '12 at 11:21

Yes, the formula has mathematical meaning. If $a$ and $b$ are natural numbers, and $a\lt b$, the binomial coefficient $\dbinom{a}{b}$ is defined to be $0$. So the sum is effectively a finite sum.

Reamrks: The convention is useful in simplifying formulas. Without it, in the formula of the post, we would have to specify that the summation is to $\min(m,n)$. In situations with more summations, the convention can make for considerable simplifications.

share|cite|improve this answer
Thank you, Andre! – Moki Oct 11 '12 at 12:09

It's quite easy to verify that your sum is equal to the coefficient of $z^m$ in the product:

$$ (1+xz)^n\,(1+z)^m.$$

If you set $x=1$ you can find a slight generalization of the Chu-Vandermonde identity:

$$ \sum_{k=0}^{+\infty}\binom{m}{k}\binom{n}{k}=\binom{m+n}{n}.$$

share|cite|improve this answer
Hi Jack, thanks for answering! – Moki Oct 11 '12 at 15:00

It's summable, as $\binom pk=0$ whenever $k\geq p$. (so in the sum there are $\min\{m,n\}+1$ non-vanishing terms.

share|cite|improve this answer
Thank you very much for your answer! – Moki Oct 11 '12 at 11:49

It is summable and has a closed form in terms of hypergeometric function

$$ F(-n,-m;\,1;\,x) \,.$$

The above hypergeometric function can be simplified to a polynomial in the following cases

1) if $n$ and $m$ are non-negative integers.

2) if $n$ or $m$ is a non-negative integer.

share|cite|improve this answer
Thanks a lot, Mhenni! – Moki Oct 11 '12 at 12:09
@Moki: You are welcome. – Mhenni Benghorbal Oct 11 '12 at 12:20

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.