The expression I am trying to simplify is the following:

$$f(x)=\sum_{n\ge k}S(n,k)(L)_{n}x^n$$ where $S(k,n)$ is the Stirling number of second kind for $k,n\in \mathbb{Z}^+,\ L\ge n$ and $(L)_n$ is the falling factorial defined as $(L)_n=L(L-1)\cdots (L-n+1)$. Note that if $(L)_n$ were replaced by $1$, then the expression becomes a well known identity yielding $$\prod_{r=1}^k\frac{x}{1-rx}$$

Please give helpful hints or direct me to literature that may help to solve this problem. Thanks in advance.

  • 1
    $\begingroup$ @MithleshUpadhyay I don't know what good does your reference do to solve my problem. I am well aware of what the Stirling number of second kind is. But the question asks how I can use that definition or some other property to solve this particular problem. So something which is a little bit more thorough might be helpful. $\endgroup$ – Samrat Mukhopadhyay Apr 10 '16 at 9:11

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