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We introduce two ordinary second-order linear differential equations of theLaguerre- and Jacobi-type. Solutions are written as infinite series of squareintegrable functions in terms of the Laguerre and Jacobi polynomials,respectively. The expansion coefficients of the series satisfy three-termrecursion relations, which are solved in terms of orthogonal polynomials withcontinuous and/or discrete spectra. Most of these are well-known polynomialswhereas few are not. We present physical applications of these differentialequations in quantum mechanics.

Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications