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Question: How to calculate this summation $S=\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k} $? Where $m<n_1,m<n_2$

Remark1: When $a=b$, I know the above summation $S=a^m\sum_{k=0}^m {n_1\choose k} {n_2\choose m-k} =a^m {n_1+n_2\choose m} $.

Remark2: This summation looks somewhat similar to the usual binomial formula $\sum_{k=0}^m a^k b^{m-k} {m\choose k} =(a+b)^m$. So is there also a similar formula for $S$?

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    $\begingroup$ It is the coefficient of $x^m$ in $(1+ax)^{n_1}(1+bx)^{n_2}$. In general there is no closed form expression without using special functions. $\endgroup$
    – Fedor Petrov
    Commented Sep 29, 2022 at 19:32
  • $\begingroup$ @Fedor Petrov Thank you! $\endgroup$
    – Dian
    Commented Sep 29, 2022 at 20:01

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In terms of a hypergeometric function you would have $$S=\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k}=b^m \binom{n_2}{m} \, _2F_1\left(-m,-n_1;-m+n_2+1;a/b\right).$$ I don't see a simpler closed-form expression for arbitrary parameters, but if you fix $n_1$ this does simplify to a simple rational function of $m,n_2,a/b$.

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