Recursion relations for solutions to the Schrodinger equation

The author considers the general eigenfunction for a second-order differential operator in one variable. For many well-known elementary functions, they can also find a three-term recursion relation in the eigenvalue parameter. For practical computation, this is a very desirable property. Examples include Legendre functions, Bessel functions, etc. In [Math. Z., 29 (1929), pp. 730-736], Bochner showed that the only polynomial solutions to this problem were the well-known ones. This paper looks for solutions that may not be polynomials. It is known that, unfortunately, for the simple recursion relation considered here, no really new examples exist