Perturbations around the zeros of classical orthogonal polynomials
Abstract
Starting from degree N solutions of a time dependent Schrödinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the polynomial is derived. The matrix has remarkable Diophantine properties. Its eigenvalues are independent of the zeros. The corresponding eigenvectors provide the representations of the lower degree (0,1,…,N−1) polynomials in terms of the zeros of the degree N polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.
- Publication:
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Journal of Mathematical Physics
- Pub Date:
- April 2015
- DOI:
- 10.1063/1.4918707
- arXiv:
- arXiv:1411.3045
- Bibcode:
- 2015JMP....56d2106S
- Keywords:
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- Mathematics - Classical Analysis and ODEs;
- High Energy Physics - Theory;
- Mathematical Physics;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- Quantum Physics
- E-Print:
- LaTeX2e 31 pages, no figure