  Using an algebraic method for solving the wave equation in quantum mechanics,
we encountered a new class of orthogonal polynomials on the real line. It
consists of a four-parameter polynomial with continuous spectrum on the whole
real line and two of its discrete versions; one with a finite spectrum and
another with countably infinite spectrum. A second class of these new
orthogonal polynomials appeared recently while solving a Heun-type equation.
Based on these results and on our recent study of the solution space of an
ordinary differential equation of the second kind with four singular points, we
introduce a modification of the Askey scheme of hyper-geometric orthogonal
polynomials. Up to now, these polynomials are defined by their three-term
recursion relations and initial values. However, their other properties like
the weight functions, generating functions, orthogonality, Rodrigues-type
formulas, etc. are yet to be derived analytically. Due to the prime
significance of these polynomials in physics and mathematics, we call upon
experts in the field of orthogonal polynomials to study them, derive their
properties and write them in closed form (e.g., in terms of hypergeometric
functions).
