Series solution of a ten-parameter second order differential equation with three regular and one irregular singularities

Abstract

We introduce a ten-parameter ordinary linear differential equation of the second order with four singular points. Three of these are finite and regular whereas the fourth is irregular at infinity. We use the tridiagonal representation approach to obtain a solution of the equation as bounded infinite series of square integrable functions that are written in terms of the Jacobi polynomials. The expansion coefficients of the series satisfy three-term recursion relation, which is solved in terms of a modified version of the continuous Hahn orthogonal polynomial.