# Properties of the Zeros of the Polynomials Belonging to the Askey Scheme

@article{Bihun2014PropertiesOT, title={Properties of the Zeros of the Polynomials Belonging to the Askey Scheme}, author={Oksana Bihun and Francesco Calogero}, journal={Letters in Mathematical Physics}, year={2014}, volume={104}, pages={1571-1588} }

In this paper, we provide properties—which are, to the best of our knowledge, new—of the zeros of the polynomials belonging to the Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted.

## 15 Citations

### Gradient system for the roots of the Askey-Wilson polynomial

- MathematicsProceedings of the American Mathematical Society
- 2019

Recently, it was observed that the roots of the Askey-Wilson polynomial are retrieved at the unique global minimum of an associated strictly convex Morse function [J. F. van Diejen and E. Emsiz,…

### Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials

- MathematicsLetters in Mathematical Physics
- 2018

Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey–Wilson, Wilson and…

### Solutions of convex Bethe Ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials

- Mathematics
- 2019

Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey–Wilson, Wilson and…

### Perturbations around the zeros of classical orthogonal polynomials

- Mathematics
- 2015

Starting from degree N solutions of a time dependent Schrodinger-like equation for classical orthogonal polynomials, a linear matrix equation describing perturbations around the N zeros of the…

### Diophantine properties of the zeros of (monic) polynomials the coefficients of which are the zeros of Hermite polynomials

- Mathematics
- 2015

We introduce a monic polynomial p_N(z) of degree N whose coefficients are the zeros of the N-th degree Hermite polynomial. Note that there are N! such different polynomials p_N(z), depending on the…

### New Properties of the Zeros of Classical and Nonclassical Orthogonal Polynomials

- Mathematics
- 2016

We identify a new class of remarkable algebraic relations satisfied by the zeros of classical and nonclassical orthogonal polynomials. Given an orthogonal polynomial family {pν(x)}ν=0, we relate the…

### New Properties of the Zeros of Krall Polynomials

- Mathematics
- 2016

We identify a new class of algebraic relations satisfied by the zeros of orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two, known as Krall…

### New Properties of the Zeros of Certain Nonclassical Orthogonal Polynomials

- Mathematics
- 2017

We propose new and remarkable algebraic identities satisfied by the zeros of certain nonclassical orthogonal polynomials. To prove the identities, we generalize the notion of spectral and…

### Inner bounds for the extreme zeros of 3F2 hypergeometric polynomials

- Mathematics
- 2017

ABSTRACT Zeilberger's celebrated algorithm finds pure recurrence relations (w.r.t. a single variable) for hypergeometric sums automatically. However, in the theory of orthogonal polynomials and…

### Solvable dynamical systems and isospectral matrices defined in terms of the zeros of orthogonal or otherwise special polynomials.

- Mathematics
- 2018

Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed.…

## References

SHOWING 1-10 OF 29 REFERENCES

### Properties of the Zeros of the Polynomials Belonging to the Askey Scheme

- Mathematics, PhilosophyLetters in Mathematical Physics
- 2014

In this paper, we provide properties—which are, to the best of our knowledge, new—of the zeros of the polynomials belonging to the Askey scheme. These findings include Diophantine relations satisfied…

### Properties of the Zeros of the Sum of two Polynomials

- Mathematics, Philosophy
- 2013

Some properties—including relations having a Diophantine character—of the zeros of the sum of two polynomials are reported.

### Tridiagonal matrices, orthogonal polynomials and Diophantine relations: II

- Mathematics
- 2007

In this paper we apply the machinery of the previous paper of this series to results reported in our first paper devoted to these topics: in particular, we extend and prove certain Diophantine…

### Properties of the zeros of the sum of three polynomials

- Mathematics
- 2013

A system of algebraic equations satisfied by the zeros of the sum of three polynomials are reported.

### Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials

- Mathematics
- 2007

Certain Diophantine conjectures are proven, and to do so certain remarkable classes of orthogonal polynomials are identified, yielding additional Diophantine findings.

### Diophantine properties of the zeros of certain Laguerre and para-Jacobi polynomials

- Mathematics
- 2012

Matrices and of arbitrary rank N, given by simple expressions in terms of the N zeros of certain Laguerre or para-Jacobi polynomials of degree N, feature a Diophantine property. In the Laguerre case,…

### Additional Recursion Relations, Factorizations, and Diophantine Properties Associated with the Polynomials of the Askey Scheme

- Mathematics
- 2009

In this paper, we apply to (almost) all the “named”
polynomials of the Askey scheme, as defined by their standard
three-term recursion relations, the machinery developed in
previous papers. For…

### On the Zeros of Polynomials Satisfying Certain Linear Second-Order ODEs Featuring Many Free Parameters

- Mathematics
- 2013

Certain techniques to obtain properties of the zeros of polynomials satisfying second-order ODEs are reviewed. The application of these techniques to the classical polynomials yields formulas which…

### Tridiagonal matrices, orthogonal polynomials and Diophantine relations: I

- Mathematics
- 2007

It is well known that the eigenvalues of tridiagonal matrices can be identified with the zeros of polynomials satisfying three-term recursion relations and being therefore members of an orthogonal…

### Orthogonal Polynomials

- Mathematics
- 2005

In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.