Thread: [SOLVED] The powers of a finite series of weighted exponentials - summation form?

Greetings,

I have a series of the form

F(x) = A_1 exp(-a_1*x) + A_2 exp(-a_2*x) + ... + A_N exp(-a_N*x)


Is there a formula for the quantity F(x)^m that is simple to evaluate?

I can do m nested loops and obtain the coefficients and exponents for F(x)^m, but this is very resource consuming if N and m are large. Typical values of N is in the 10s or 100s, m can be as high as 25.

I am looking for something similar to the bionomial sum.

Thank you in advance.

To clarify,

F(x)^m = SUM C(n1, n2, ..., nN) A_1^n1 A2^n2 ... A_N^nN * exp(-x(n1*a1+n2*a2+...+ nN*aN))

where the sum is carrier over ALL non-negative integers n1, n2, ..., nN such that n1+n2+...+nN = m. The above is a multinomial series.

My question, for me to find all the combinations of n1, n2, ..., nN that add to m and then evelate the term, I could use N nested loops where each runs from 0 to m. I will check every combination and select the ones that satisfy n1+n2+...+nN = m. However, this is prohibitive for large N and/or m.

Is there another way for efficient evaluation of the summation?

Thank you in advance.