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Prove that for $x,y$ positive integers,

$$\sum_{k=0}^{y-1}(-1)^k\frac{\binom{y-1}{k}}{k+x}=\frac{(x-1)!(y-1)!}{(x+y-1)!}$$

One way is to use the beta-gamma functions relation: http://en.wikipedia.org/wiki/Beta_function

Is there a direct way to calculate the sum?

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  • I do not see your formula in the link. – Lord Soth Jul 27 '13 at 18:37
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    The formula follows from the beta function by integrating the following between $u=0$ and $u=1$, $$\sum _{k=0}^{y-1}{y-1\choose k} \left( -1 \right) ^{k}{u}^{k+x-1}= \left( -u+1 \right) ^{y-1}{u}^{x-1}$$ (No idea how else to get it...) – Graham Hesketh Jul 27 '13 at 18:50
  • Gosper's algorithm (http://en.wikipedia.org/wiki/Gosper's_algorithm) is an elementary way to evaluate this sum. You can read about it in this (free) book (http://www.math.upenn.edu/~wilf/Downld.html) – OR. Jul 27 '13 at 19:48

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