Calculating a Finite Series: Finding Symmetry and Inductive Formulas

[Latrace]

Hello,

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

$\displaystyle\sum_{i=1}^{n-1}i^{\alpha}(n-i)^{\beta}.$

Thanks in advance,
Latrace

 

[micromass]

What did you already try to solve this problem?

 

[Latrace]

($n \geq 2$, of course) I tried to find an inductive formula by setting $n = 2, n = 3$ and $n = 4$, but don't find anything interesting. Of course we already knew that the thing is symmetric, symbolically it is also $\displaystyle\sum_{i=1}^{n-1}i^{\beta}(n-i)^{\alpha}$, 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 $n$, but that's $\sim (n-1)^{\beta} + (n-1)^{\alpha}$. I came across this when I wanted to calculate $\displaystyle\int_{0}^{1}x^m \mathrm{d}x$ for $m \geq 1$ explicitally using the Riemann sum.