0
votes
1answer
226 views

Proof by induction for Stirling Numbers

I am asked this: For any real number x and positive integer k, define the notation [x,k] by the recursion [x,k+1] = (x-k) [x,k] and [x,1] = x. If n is any positive integer, one can now ...
2
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1answer
402 views

Generating function for Stirling numbers, induction

I want to prove $\displaystyle\sum_{n=k}^{+\infty} \left[\begin{array}{c}n\\k\end{array}\right]\frac{z^n}{n!}=\frac{\log^k(1/(1-z))}{k!}$ by induction. For $k=0$ or $k=1$ it works, suppose it works ...