I am trying to find an analytical solution of an integral of the form $$ \int_a^{\infty} x^p [e^{\epsilon(x^{-m} - x^{-n})}-1]dx $$ where $0<a<+\infty$ and $p\in[0, 1, 2, 3]$
I understand that there is no way to express this integral using standard mathematical functions and that is fine. That said...
Is it possible to use a series expansion for the exponential term?
The term in square brackets can be expressed as: $$ \sum_{k=1}^\infty \frac{[\epsilon(x^{-m} - x^{-n})]^k}{k!} $$
but when I do this, I get an integral that diverges... I am most likely doing something wrong because I have computed the numerical solution of this integral and it does not diverge.
NOTE:
In the case where $m=2n$, we can use the exponential generating function of Hermite polynomials $H_n(x)$ : $$ \exp (2xt-t^2) = \sum_{n=0}^\infty H_n(x) \frac {t^n}{n!} $$