The problome is rewriten here: $\sum_{i=0}^k \frac{((i-k)a)^i b^{k-i}}{i!}$ where $0<a<1$, k is an integer larger than 1.
I came to this equation when i try to find some probability. I have tried some formulas on permutation and combination, fractional, but with little improvement.
I hope you can give me some sugestions! Thanks a lot!