I have an issue about finding expression involving finite summation for $_1F_1(mn,n,\delta x)$ where $m$ and $n$ are integer and $m,n>1$ for my case. I found an expression in [T.o.I. 3.383.$1^{11}$]
$$ \int_0^u x^{v-1}(u-x)^{\mu-1}e^{\beta x}dx = \mathcal{B}(\mu,v)u^{\mu+v-1} {_1F_1}(v,\mu+v,\beta u), \quad Real\left\{\mu,u\right\}>0$$ where $\mathcal{B}(.,.)$ is well-known Beta function. It seemed quite useful in the beginning, assuming $\mu$ is an integer and applying the binomial expansion to the $(u-x)^{\mu-1}$ term, I thought that I was gonna obtain what I seek. However, when I apply my case, $(m>1,n \in Z)$ violates the assumption $ Real\left\{\mu,u\right\}>0$. Briefly, have you got any idea to express $_1F_1(mn,n,\delta x)$ when $m,n>0 \in Z$ ?