Abstract "hypergeometric" orthogonal polynomials

2014/01/27 by Alexei Zhedanov, Zhedanov, Alexei
Mathematics · Physics and Astronomy · #33C45 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.1401.6754

openalex publication_date 2014/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We find all polynomials solutions P_n(x) of the abstract "hypergeometric" equation L P_n(x) = λ_n P_n(x), where L is a linear operator sending any polynomial of degree n to a polynomial of the same degree with the property that L is two-diagonal in the monomial basis, i.e. L xⁿ = λ_n xⁿ + μ_n x^n-1 with arbitrary nonzero coefficients λ_n, μ_n . Under obvious nondegenerate conditions, the polynomial eigensolutions L P_n(x) = λ_n P_n(x) are unique. The main result of the paper is a classification of all \it orthogonal polynomials P_n(x) of such type, i.e. P_n(x) are assumed to be orthogonal with respect to a nondegenerate linear functional σ. We show that the only solutions are: Jacobi, Laguerre (correspondingly little q-Jacobi and little q-Laguerre and other special and degenerate cases), Bessel and little -1 Jacobi polynomials.

Citations

• Two linear transformations each tridiagonal with respect to an eigenbasis of the other

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