Open problem in orthogonal polynomials

Abstract

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).

7 pages, 1 table, 17 references

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Cited by in corpus (14)

• Tridiagonal Representation Approach in Quantum Mechanics
• Quantum mechanics with orthogonal polynomials
• Series solutions of Heun-type equation in terms of orthogonal polynomials
• Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications
• Deformed Morse-like potential
• Bound states and the potential parameter spectrum
• Series solutions of Bessel-type differential equation in terms of orthogonal polynomials and physical applications
• Bound-states for generalized trigonometric and hyperbolic Pöschl-Teller potentials
• Mapping Schrödinger equation into a Heun-type and identifying the corresponding potential function, energy and wavefunction
• Bound-state solutions of the Schrödinger equation for two novel potentials
• Finite series representation for the bound states of a spiked isotropic oscillator with inverse-quartic singularity
• Electric dipole and quadrupole contributions to valence electron binding in a charge-screening environment
• Electrostatic multipole contributions to the binding energy of electrons
• Progressive approximation of bound states by finite series of square-integrable functions