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Perturbations around the zeros of classical orthogonal polynomials

### Abstract

Starting from degree
solutions of a time dependent Schrödinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the
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 (
) polynomials in terms of the zeros of the degree
polynomial. The results are valid universally for all the classical orthogonal polynomials, including the Askey scheme of hypergeometric orthogonal polynomials and its q-analogues.

© 2015 AIP Publishing LLC

Received 19 November 2014
Accepted 09 April 2015
Published online 23 April 2015

Acknowledgments:
It is a pleasure to thank the organizers of the CRM-ICMAT Workshop on “Exceptional orthogonal polynomials and exact solutions in mathematical physics” (Segovia, Spain, 7–12 July 2014). R.S. thanks Francesco Calogero and Kazuhiko Aomoto for useful and insightful discussion and comments. He thanks Pauchy Hwang and Department of Physics, National Taiwan University for hospitality.

Article outline:

I. INTRODUCTION
II. TIME DEPENDENT SCHRÖDINGER EQUATIONS
A. Polynomial solutions
III. PERTURBATIONS AROUND THE ZEROS
IV. EXAMPLES FROM ORDINARY QUANTUM MECHANICS
A. Hermite
B. Laguerre
C. Jacobi
V. EXAMPLES FROM DISCRETE QUANTUM MECHANICS WITH PURE IMAGINARY SHIFTS
A. Polynomials having *η*(*x*) = *x*, −∞ < *x* < ∞, *γ* = 1
1. Continuous Hahn
2. Meixner-Pollaczek
B. Polynomials having *η*(*x*) = *x*
^{2}, 0 < *x* < ∞, *γ* = 1
1. Wilson
2. Continuous dual Hahn
C. Polynomials having *η*(*x*) = cos*x*, 0 < *x* < *π*, *e*
^{
γ
} = *q*
1. Askey-Wilson
2. Continuous dual *q*-Hahn
3. Al-Salam-Chihara
4. Continuous big *q*-Hermite
5. Continuous *q*-Hermite
6. Continuous *q*-Jacobi
7. Continuous *q*-Laguerre
VI. EXAMPLES FROM DISCRETE QUANTUM MECHANICS WITH REAL SHIFTS
A. Polynomials having *η*(*x*) = *x*, [0, 1, …, *N*], or [0, 1, …, ∞)
1. Hahn
2. Krawtchouk
3. Meixner
4. Charlier
B. Polynomials having *η*(*x*) quadratic in *x*
1. Racah
2. Dual Hahn
C. Polynomials having *η*(*x*) linear in *q*
^{−x
}, [0, 1, …, *N*], or [0, 1, …, ∞)
1. *q*-Hahn
2. Quantum *q*-Krawtchouk
3. *q*-Krawtchouk
4. Affine *q*-Krawtchouk
5. *q*-Meixner
6. Al-Salam-Carlitz II
7. *q*-Charlier
D. Polynomials having *η*(*x*) linear in *q*^{x}
, [0, 1, …, ∞)
1. Little *q*-Jacobi
2. Little *q*-Laguerre/Wall
3. Alternative *q*-Charlier
E. Polynomials having *η*(*x*) bilinear in *q*
^{−x
} and *q*^{x}
, [0, 1, …, *N*]
1. *q*-Racah
2. Dual *q*-Hahn
VII. SUMMARY AND COMMENTS

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