Special Orthogonal Polynomials in Quantum Mechanics

Abstract

Using an algebraic method for solving the wave equation in quantum mechanics, we 8 encountered a new class of orthogonal polynomials on the real line. One of these is a four-parameter 9 polynomial with a discrete spectrum. Another that appeared while solving a Heun-type equation 10 has a mix of continuous and discrete spectra. Based on these results and on our recent study of the 11 solution space of an ordinary differential equation of the second kind with four singular points, we 12 introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of 13 these polynomials are defined only by their three-term recursion relations and initial values. 14 However, their other properties like the weight function, generating function, orthogonality, 15 Rodrigues-type formula, etc. are yet to be derived analytically. This is an open problem in 16 orthogonal polynomials.