integrating modified gamma functions

Question

I have an integral that looks like an incomplete gamma function, with another factor in front \begin{equation} I = \int_x^y \frac{1}{a^{c+1}}[1-t]^bt^ce^{-\frac{t}{a^2}}dt \end{equation} Is there a way to evaluate an integral of this form? In this case $c\in \mathbb Z$ and $b,a\in \

Answer 1

Note that $$(1-t)^bt^c = \sum_{i=0}^b\begin{pmatrix}b\\i\end{pmatrix}(-1)^{i}t^{c+i}=\sum_i d_it^{c+i}$$ Then we can evaluate the integral of the sum as the sum of the integrals

$$\sum_i\int_x^y\frac{d_i}{a^{c+1}}t^{c+i}e^{-\frac{t}{a^2}}dt $$

Which can be integrated using (incomplete) $\Gamma$ functions.