Finding z transform of this function.

Question

I need to find the z-transform of the function $t^{a}e^{-bt}$. Though it did appear easy to me initially but i cannot figure out how to do so. The problem is that 'a' can be a non-integer. (t can be thought of as a discrete time variable) Can someone please guide me or give some hints. Thanks in advance !!

Answer 1

I guess we are dealing with one-side transform here. Then (assuming $a>0$) $$F(z)=\sum_{t=0}^{\infty} t^a e^{-bt} z^{-t}=\sum_{t=1}^{\infty} \frac{w^{t}}{t^{-a}}={\rm Li}_{-a}(w)$$ where $w=z^{-1}\,e^{-b}$ and ${\rm Li}_s(\cdot)$ is the polylogarithm function. This has no nice simple closed form expression for arbitrary $a$. Is this howework? Are you sure you got it right?