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?



Generalized multinomial coefficient
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. 


