I've been dealing with some problem and faced a necessity to obtain the eigenproblem for the following second-order differential operator \begin{equation} H = x(Ax-g)\frac{d^2}{dx^2} + x(A(1-2j) + B) \frac{d}{dx} + 2gjx + j(Aj-B), \end{equation} where $A,B,G$ - some real constants, and $j\in \mathbb{N}$.
The eigenproblem reads $$ H \psi = E \psi, $$ which can be rewritten (assuming $x \neq 0, g$) in the form of a second-order differential equation $$ \psi'' + \frac{\alpha}{x-\sigma} \psi' + \frac{\beta x + \gamma}{x(x-\sigma)} \psi = 0, $$ where I denoted $$ \sigma = g/A, \quad \alpha = (1-2j) + B/A, \quad \beta = 2gj/A, \quad \gamma = j(j-B/A) - E/A. $$
I am not very deep into special functions. So, before looking for the solution in the form of a series or generalised series, I would like to ask a question here.
Could you tell me please if this equation is related or equivalent to some known one?