A problem I am trying to tackle seems like it could be significantly simplified if it were possible to choose a basis for the relevant function space as something like:
$$ \mathcal{B}_\mathrm{B} = \left\{ x^\alpha (1-x)^\beta \; : \; \alpha \in \mathbf{R}, \, \beta \in \mathbf{R}\setminus\mathbf{N} \right\} $$
or some variation of the above. I have chosen the restriction on $\beta$ to remove the obvious linear-dependence relation otherwise.
I know the literature is full of various bases for different function spaces, e.g. arising from Sturm-Liouville problems, but don't recall seeing anything like this and couldn't find it in the references I consulted.
Does a basis of this form exist in the literature, does it have a name I should be aware of, and is there an integral kernel with respect to which these functions are orthogonal/orthonormal?
If not, is there an easy way to try to find one?