While solving a problem I started to compute a Taylor series for reciprocal of a polynomial $f(x)$: $$ f(x)=a_0+a_1x+a_2x^2+\cdots;\quad 1/f (x)=c_0+c_1x+c_2x^2+\cdots, $$ with $a_0\ne0$.
After computing several terms of the latter expansion I realized the general pattern: $$ c_n= \sum_{p_k\ge0}^{\sum k p_k=n}(-1)^p\binom {p}{p_1;p_2;\cdots}a_0^{-(p+1)}\prod_k a_k^{p_k},\tag1 $$ where $p=\sum_k p_k $.
From comparison of (1) with Taylor expansion of $[f(x)]^N $ (with $N$ being a positive integer): $$ C_n=\sum_{p_k\ge0}^{\sum k p_k=n}\binom {N}{p_0;p_1;p_2;\cdots}\prod_k a_k^{p_k}\tag2 $$ it is tempting to identify: $$(-1)^p\binom {p}{p_1;p_2;\cdots}=\binom {-1}{p_0;p_1;p_2;\cdots}.\tag3 $$ There is however a subtle distinction between (1) and (2). Whereas the summation in the former runs only over positive $k $, in the latter it includes also $k=0$.
After some initial doubts I came to conclusion that in fact everything is fine and can be reduced to the general definition of multinomial: $$ \binom {p}{p_0;p_1;p_2;\cdots}=\frac{p!}{p_0!p_1!p_2!\cdots}, $$ where $p$ is assumed to be the sum of lower indicies. As well-known for positive $p$ the multinomial degenerates to $0$ when at least one of $p_k$ is negative. It behaves however quite differently if $p$ is negative. Then the result is finite if a single $p_k$ is negative. There is an unpleasent problem of a quotient of two infinities, but formal writing $$ \frac{(-1)!}{(-p-1)!}:=\lim_{z\to0}\frac{\Gamma(z)}{\Gamma(z-p)}=(-1)^pp! $$ looks quite convincing and allows immediate generalization of (3) to: $$\binom {-N}{p_0;p_1;p_2;\cdots}=(-1)^p\binom {p+N-1}{N-1;p_1;p_2;\cdots},\tag4 $$ where $N$ is a positive integer and $p_0=-N-p$ is assumed. Observe that (4) gives correct result for the well-known expression for binomial of negative powers.
Do the considerations make sense? Can anybody give a reference to a rigorous derivation? I looked through MSE archives and found that the question of multinomial theorem for negative powers was raised several times but never received an answer. From practical point of view it would be nice for me to let the equality (1) be a simple corollary of a "general" multinomial theorem (cf. (2)) without a need for a separate proof.