Open Problem in Orthogonal Polynomials.

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered new families of orthogonal polynomials on the real line. The properties of the physical system (e.g. energy spectrum, phase shift, density of states, etc.) are obtained from the properties of these polynomials. One of these new families is composed of four-parameter polynomials describing a discrete spectrum of the corresponding quantum mechanical system. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their three-term recursion relations and initial values. However, their other properties like weight functions, generating functions, orthogonality, Rodrigues-type formula, etc., are yet to be derived analytically. Obtaining these properties is an open problem in orthogonal polynomials. [ABSTRACT FROM AUTHOR]