Calculating a finite series

Latrace

Hello,

I would love some help on calculating the following sum for [itex]\alpha, \beta \in \mathbb{N}[/itex] and [itex]n \in \mathbb{N} \backslash \{0\}[/itex]:

[itex]\displaystyle\sum_{i=1}^{n-1}i^{\alpha}(n-i)^{\beta}.[/itex]

Thanks in advance,
Latrace

micromass

What did you already try to solve this problem?

Latrace

([itex] n \geq 2 [/itex], of course) I tried to find an inductive formula by setting [itex] n = 2, n = 3 [/itex] and [itex] n = 4 [/itex], but don't find anything interesting. Of course we already knew that the thing is symmetric, symbolically it is also [itex] \displaystyle\sum_{i=1}^{n-1}i^{\beta}(n-i)^{\alpha} [/itex], but that's about all I find when I try to find an inductive formula. I think now that this might be the easiest way to express the series.
What I eventually need is the behavior for large [itex] n [/itex], but thats [itex] \sim (n-1)^{\beta} + (n-1)^{\alpha} [/itex]. I came across this when I wanted to calculate [itex] \displaystyle\int_{0}^{1}x^m \mathrm{d}x [/itex] for [itex] m \geq 1 [/itex] explicitally using the Riemann sum.