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

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 threeterm 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.MSC: 34-xx, 81Qxx, 33C45, 33D45

Series solutions of Heun-type equation in terms of orthogonal polynomials

“…x making the overall multiplicative factor 1 ( 1) (i.e., taking care of the normalization factor n c ). Requiring symmetry of the tridiagonal representation (i.e., the n-dependent factor multiplying 1 ( )…”

“…where   0 1 , , , , a b A A A  are real parameters. We obtained their solutions as infinite series in terms of orthogonal polynomials with continuous and/or discrete spectra [1]. Most of those are well-known polynomials while few are not.…”

Series Solution of a Ten-Parameter Second-Order Differential Equation with Three Regular Singularities and One Irregular Singularity

“…Recently, we introduced the following hypergeometric-type and confluent-hypergeometrictype differential equations     A a discrete parameter (quadratic and linear in an integer n, respectively). We obtained solutions of these two equations as infinite or finite series of square integrable functions with the expansion coefficients being orthogonal polynomials of continuous and/or discrete spectra [1]. Most of those are well-known polynomials while few are not.…”

Tridiagonal representation approach in quantum mechanics

“…As examples, we mention the anion problem where an electron becomes bound to a neutral molecule with an electric dipole moment [16,17], the binding of a charged particle to an electric quadrupole in two dimensions [18], energy density bands engineering [19], electric dipole and quadrupole contributions to valence electron binding in a charge-screening environment [20], etc. In applied mathematics, the TRA was also used to obtain series solutions of new types of ordinary differential equations of the second order with three and four singular points [21][22][23]. In the following section, we start by introducing and formulating the TRA and explain its two working modes.…”

“…The four dimensionless real parameters are such that   greater than 1. Using the differential equation and differential property of the Jacobi polynomial, we obtain the following matrix representation of the wave operator in this basis[21,31] U and U  are arbitrary dimensionless parameters. The relevant case to the present treatment is the first one corresponding to…”