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.