Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications
arXiv:1802.09708 · doi:10.1063/1.5027158
Abstract
We introduce two ordinary second-order linear differential equations of the Laguerre- and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Most of these are well-known polynomials whereas few are not. We present physical applications of these differential equations in quantum mechanics.
28 pages, 5 sections, 2 Appendices, 24 references
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- Bound-states for generalized trigonometric and hyperbolic Pöschl-Teller potentials
- Solutions of the scattering problem in a complete set of Bessel functions with a discrete index
- Structural Algebraic Quantum Field Theory
- Progressive approximation of bound states by finite series of square-integrable functions