A fractional order transfer function looks like this
$$ \frac{Y}{U}=\frac{b_ns^{m\alpha}+\dots+b_1s^{\alpha}+b_0}{a_ns^{n\alpha}+\dots+a_1s^{\alpha}+a_0} $$
where $\alpha \in (0,1)$ and it is often a rational number.
But I am wondering how does their discrete format look like.
I suspect if the normal discrete system is $$ y(t+t_s)=f(u(t)) $$ where $t_s$ is a sample time, then a fractional order transfer function has dependence is like $y(t+\alpha t_s)=f_2(u(t))$ but it cannot be true because this looks like a first order but with a different sample time. So, how do discrete form of fractional order system look like and how are they different?
To make this question clear and avoid an open discussion, what is the discrete form of this single order transfer function in time domain?
$$ \frac{Y}{U}=\frac1{s^\alpha+a_0} $$