How compute $\sum_{j=1}^k \binom{x}{j}\binom{k-1}{j-1}\alpha^j, \quad x, \alpha\in\mathbb{R}$
1 Answer
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The sum equals $x \, _2F_1(1-k,1-x;2;a).$ For more on this, check out Abramowitz and Stegun, or some such.