Say say you have a binomial expansion of a product like $(1+ax)^n * (1+bx)^m$, I want an explicit formula which gives the coefficients of the polynomial you get when you expand the product out. So here is what I have come up with.
$Cj = \sum_{k=0}^{k=j} a^k b^{j-k} * \binom{n}{k} * \binom{m}{j-k}$
I have this expression, and it looks something like the vandermonde identity but it is not quite the same: can it be simplified?
What I've considered:
Writing a maclaurain series, however the problem of this approach is that derivatives get ugly
$$C_j=b^{j} \sum_{k=0}^{j} {n \choose k} {m \choose j-k}\left(\frac{a}{b}\right)^k.$$ The required coefficient $$C_j=b^{j} {n \choose j} ~_2F_1(-m,-j;-1+n-j;\frac{a}{b})$$ Where $$~_2F_1(a,b;c;z)=1+\frac{a b}{c}\frac{x}{1!}+\frac{a(a+1)b(b+1)}{c(c+1) }\frac{x^2}{2!}+\frac{a(a+1)(a+2)b(b+1)(b+2)}{c(c+1)(c+2)}\frac{x^3}{3!}.....$$