The solutions to the Jacobi differential equation: $\left(1-x^{2}\right)y''(x)+(\beta -\alpha -(\alpha +\beta +2)x)y'(x)+n(n+\alpha +\beta +1)y(x)=0,$ are hyper-geometric functions
$\,{}_{2}F_{1}\left(-n,1+\alpha +\beta +n;\alpha +1;{\tfrac {1}{2}}(1-x)\right),$
Imposing boundary conditions in the region $(-1,1)$, we get the Jacobi polynomials, namely, these functions (in some normalization) with $n$ positive integer.
From now on let us specify to $\beta=0$ for simplicity.
However, if one wants to span the relevant space of functions on $(1,\infty)$, one ends up with the family of functions with $n=-\frac{\alpha+1}{2}+i\rho$ with real $\rho$.
In principle, orthogonality with the weight function $(1-x)^{\alpha}$ follows from the fact that these are eigen-functions with different eigenvalues and appropriate boundary conditions.
My questions are (answers to any would be great!):
1) How does one find the normalization? Doing the integrals explicitly is very hard even numerically (the integrand decays only as $1/x$ but is oscillatory)
2) Is there a way to establish the space these functions span as a function of $\rho$? (Maybe the most important part of the question for me)
3) Should there be an orthogonality relation also in $\rho$ space?
Thanks in advance!!