I am looking for a closed-form expression for the finite sum of the product of an exponential function with a polynomial function --- that is, the sum
N ∑ a^n * n^k n=0
where N and k (but not a) are non-negative integers.
Now, for my problem I don't necessarily need the exact answer as long as I can know that it is always an exponential polynomial (i.e., a product of polynomial and exponential functions). I can see that this is true for the values of k=0...3 (thanks to Wolfram Alpha), but I can't see enough in the pattern to make a good guess at the general answer.
If I ask Wolfram Alpha to calculate the above sum then it gives me:
sum_(n=0)^N a^n n^k = Li_(-k)(a)-a^(N+1) Phi(a, -k, N+1)
where Li is the "polylogarithmic" function and Phi is the "Lerch transcendental function", but this not very helpful because I am not interested in the full analytic case where k could be an arbitrary real or complex number but rather the case where it is a non-negative integer. It is hard for me to see from the above whether for integer values of k it reduces to an exponential polynomial.
So in short, does anyone where I could find a useful closed form for the sum above, or at least where I could find out whether the sum results in an exponential polynomial?