I have an integral of the form
$$ \int_{0}^{\infty} x^{\mu-1}e^{-\beta x} \Gamma\left( \nu,\alpha x \right)dx$$
where $\Gamma\left(s,x\right) = \int_{x}^{\infty}t^{s-1}e^{-t}$ is the upper incomplete gamma function. Using the answer provided by Table of Integrals, Series, and Products by Daniel Zwillinger (equation 6.455 page 657), the solution to this integral can be written as
$$\frac{\alpha^\nu \Gamma\left( \mu+\nu \right)}{\mu \left(\alpha + \beta\right)^{\mu+\nu}}{}_2F_1\left( 1,\mu+\nu;\mu+1;\frac{\alpha}{\alpha+\beta} \right) $$
where ${}_2F_1$ is the hypergeometric function. Unfortunately, in my case the parameters are defined as $\mu = m, \nu = -m, \alpha = \frac{m}{\omega}, \beta = -\frac{m}{\omega}$ which makes the solution invalid! What should I do in this case? BTW I get a totally different solution by Mathematica which is written in terms of Polygamma function (It also becomes invalid). What should I do?!