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3 votes
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How to calculate this limit (if exist)?

$\newcommand\ka\kappa$The problem obviously reduces to this one: find \begin{equation} L:=\lim_{m\to\infty}\frac{S(c,m-1,n_1-1,n_2-1)}{S(c,m,n_1,n_2)}, \end{equation} where $c:=a/b\ne1$ and \begin{...
5 votes
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How to calculate this summation $\sum_{k=0}^m a^k b^{m-k} {n_1\choose k} {n_2\choose m-k} $?

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 ...
2 votes

Closed expression for hypergeometric sum

The Mathematica formula mentioned by Rivin is derived via \begin{equation} \sum_{m=0}^l \binom{s-m}{s-l}\binom{s-1+m}{s-1}x^m = \sum_{m=0}^l \frac{\Gamma(s-m+1)\Gamma(s+m)}{\Gamma(1+s-l)\Gamma(l+1-m)\...

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