Thanks, dear @Carl Love, it's a brilliant solution.
I want to change the question a little bit.
Let's select as follows:
is a real number.
In fact; it is a generalization of the question in the post. If we select , it transforms to original post.
How to get an explicit formula?
I wrote a code as follows by adapting your code above:
Although I waited a lot, Maple did not give any results. (I think we need more assumptions )
assume(n::posint, j::nonnegint, j <= n, T > 0, alpha > 0,rho::realcons);
local i:= op(procname);
if i=0 then piecewise(t<0, 0, t<h, 1-t/h)
elif i=n then piecewise(t < T-h, 0, 1-(T-t)/h)
t < (i-1)*h, 0,
t < i*h, t/h-(i-1),
t < (i+1)*h, (i+1)-t/h
int(t^(rho)*((j*h)^(rho+1)-t^(rho+1))^(alpha-1)*psi[i](t), t= 0..j*h)*(rho+1)^(1-alpha)/GAMMA(alpha)
g[0,j]() assuming additionally, j > 0 ;
g[j,j]() assuming additionally, j > 0;
g[i,j]() assuming i::posint, additionally, i < j;