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

The usual binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!} $ can be generalized to real upper argument, lower argument still a nonnegative integer, by the definition $\binom{\alpha}{k} = \frac{\alpha (\alpha-1)\dots (\alpha-k+1)}{k!}$.

In the same way we could generalize the multinomial coefficient $ \binom{k_1 + \dots + k_s}{k_1 k_2 \dots k_s} = \frac{(k_1+\dots+k_s)!}{k_1! k_2! \dots k_s!}$ to a real upper argument by $ \binom{\alpha}{k_1 k_2 \dots k_s} = \frac{\alpha (\alpha-1) \dots (\alpha -k_1 -\dots -k_s+1)}{k_1! \dots k_s!}$ .

This definition, for instance, will allow us to generalize the Newton Binomial Theorem, to "the Newton Multinomial Theorem", in the following way (trinomial case):

\begin{equation} (1+y+z)^{\alpha} = \sum_{r=0}^\infty \sum_{s=0}^\infty \binom{\alpha}{r s} y^r z^s \end{equation}

My question is simple: I did a lot of googling, but I cannot find any reference for this!

¿Any comments? ¿Any references?

share|cite|improve this question
up vote 3 down vote accepted

We have $$(1+y+z)^{\alpha} = \sum_{t=0}^{\infty} \binom{\alpha}{t}(y+z)^t$$ The trinomial identity is just a rearrangement of the above by $$\binom{\alpha}{r,s}=\binom{\alpha}{r+s}\binom{r+s}{s}$$

To have convergence, we assume $|y+z|<1$, $|y|<1$, and $|z|<1$. The generalization follows similarly.

share|cite|improve this answer
Yes, but the question was to references (published! or web-pages) for the generalization of the multinomial coefficient. – kjetil b halvorsen May 4 '12 at 4:13
I don't see any reason that anyone would want to publish this. I think it will be okay to just use this fact without proof. – i707107 May 4 '12 at 4:29

I don't see a reference but the point of the answer above is that your generalized multinomial coefficient is always the product of a generalized binomial coefficient and an ordinary multinomial coefficient. $$\binom{\alpha}{k_1 k_2 \dots k_s} =\binom{\alpha}{k}\binom{k}{k_1 k_2 \dots k_s}$$ for $k=k_1+k_2+\cdots k_s.$

The general binomial theorem is usualy derived as a very special Taylor series. In the same way the multivariate Taylor series for $(x_1+x_2+\cdots+x_s)^{\alpha}$ comes out to be just what you want.

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.