I was reading a paper and was unable to follow a step (getting to equation 16 from equation 15)
$$ p(q,s,t) = \frac{\Gamma(am + q)}{\Gamma(am) q!} \left(\frac{s}{t} \right)^\frac{am}{1+a} \left[1 - \left(\frac{s}{t}\right)^\frac{1}{1+a}\right]^q $$
I want to show that the average over $q$ that is,
$$\sum_{q=0}^{\infty} q p(q,s,t) = am \left[ \left(\frac{s}{t} \right)^{-\frac{1}{1+a}} - 1\right] $$
I tried simplifying the ratio of gamma functions in terms of factorials but that didn't help with the sum, can someone help me prove this or point me in the right direction?