# Bounds for Extreme Zeros of Quasi-orthogonal Ultraspherical Polynomials

@article{Driver2016BoundsFE, title={Bounds for Extreme Zeros of Quasi-orthogonal Ultraspherical Polynomials}, author={Kathy Driver and Martin E. Muldoon}, journal={arXiv: Classical Analysis and ODEs}, year={2016} }

We discuss and compare upper and lower bounds obtained by two different methods for the positive zero of the ultraspherical polynomial $C_{n}^{(\lambda)}$ that is greater than $1$ when $-3/2 < \lambda < -1/2.$ Our first approach uses mixed three term recurrence relations and interlacing of zeros while the second approach uses a method going back to Euler and Rayleigh and already applied to Bessel functions and Laguerre and $q$-Laguerre polynomials. We use the bounds obtained by the second… Expand

#### Figures from this paper

#### 6 Citations

Zeros of quasi-orthogonal ultraspherical polynomials

- Mathematics
- 2016

Abstract For each fixed value of λ in the range − 3 / 2 λ − 1 / 2 , we prove interlacing properties for the zeros of polynomials, of consecutive and non-consecutive degree, within the sequence of… Expand

Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomials

- Mathematics, Computer Science
- Numerical Algorithms
- 2020

An algorithm for generating infinite monic orthogonal sequences generated by applying Wendroff’s Theorem to the interlacing zeros of C n − 1 λ ( x) and − 3/2 < λ < −’ 1/2, λ ≠ −1,…. Expand

On quasi-orthogonal polynomials: Their differential equations, discriminants and electrostatics

- Mathematics
- Journal of Mathematical Analysis and Applications
- 2019

Abstract In this paper, we develop a general theory of quasi-orthogonal polynomials. We first derive three-term recurrence relation and second-order differential equations for quasi-orthogonal… Expand

Zeros of Jacobi polynomials Pn(α,β)$ P_{n}^{(\alpha ,\beta)} $, − 2 < α, β < − 1

- Mathematics
- Numerical Algorithms
- 2018

The sequence of Jacobi polynomials {Pn(α,β)}n=0∞$\{P_{n}^{(\alpha ,\beta )}\}_{n = 0}^{\infty }$ is orthogonal on (− 1,1) with respect to the weight function (1 − x)α(1 + x)β provided α > − 1,β > −… Expand

Algorithmic Methods for Mixed Recurrence Equations, Zeros of Classical Orthogonal Polynomials and Classical Orthogonal Polynomial Solutions of Three-Term Recurrence Equations

- Mathematics
- 2019

v 0 General Introduction 1 1 Preliminary results 6 1.1 Interlacing properties for zeros of sequences of classical orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .… Expand

Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials

- Mathematics
- 2018

This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14).

#### References

SHOWING 1-10 OF 31 REFERENCES

Interlacing Properties and Bounds for Zeros of Some Quasi-Orthogonal Laguerre Polynomials

- Mathematics
- 2015

We discuss interlacing properties of zeros of Laguerre polynomials of different degree in quasi-orthogonal sequences $$\{L_{n}^{(\alpha )}\} _{n=0}^\infty $${Ln(α)}n=0∞ characterized by $$-2<\alpha… Expand

Zeros of quasi-orthogonal ultraspherical polynomials

- Mathematics
- 2016

Abstract For each fixed value of λ in the range − 3 / 2 λ − 1 / 2 , we prove interlacing properties for the zeros of polynomials, of consecutive and non-consecutive degree, within the sequence of… Expand

Interlacing of zeros of Gegenbauer polynomials of non-consecutive degree from different sequences

- Mathematics, Computer Science
- Numerische Mathematik
- 2012

It is proved that Stieltjes interlacing holds between the zeros of the kth derivative of $${C_{n}^{\lambda}}$$ and theZeros of C_{n+1}^(\lambda)$, and associated polynomials are derived that play an analogous role to the de Boor–Saff polynmials in completing the interlaces process of the zoes. Expand

Bounds for the small real and purely imaginary zeros of Bessel and related functions

- Mathematics
- 1995

We give two distinct approaches to finding bounds, as functions of the order ν, for the smallest real or purely imaginary zero of Bessel and some related functions. One approach is based on an old… Expand

Zeros of ultraspherical polynomials and the Hilbert-Klein formulas

- Mathematics
- 2001

Abstract The orthogonality of the ultraspherical polynomials Cnλ(z) for λ>− 1 2 ensures that all of their zeros are in the interval (−1,1). In a previous paper (Driver and Duren, Indag. Math. 11… Expand

Trajectories of the zeros of hypergeometric polynomials F(−n, b; 2b; z) for b < − 1/2

- Mathematics
- 2001

In a previous paper [2] we studied the zeros of hypergeometric polynomials F(−n, b; 2b; z), where b is a real parameter. Making connections with ultraspherical polynomials, we showed that for b > −… Expand

Interlacing of zeros of orthogonal polynomials under modification of the measure

- Computer Science, Mathematics
- J. Approx. Theory
- 2013

It is proved that the zeros of these polynomials, if they are of equal or consecutive degrees, interlace when either 0.<@t,@c@?1 or @c=0 and 0<@t@?2. Expand

On quasi-orthogonal polynomials

- Mathematics
- 1990

Abstract Chihara [On quasi orthogonal polynomials, Proc. Amer. Math. Soc. 8 (1957) , 765–767] has shown that quasi-orthogonal polynomials satisfy a three-term recurrence relation with polynomial… Expand

INEQUALITIES FOR THE SMALLEST ZEROS OF LAGUERRE POLYNOMIALS AND THEIR q-ANALOGUES

- 2007

We present bounds and approximations for the smallest positive zero of the Laguerre polynomial L n (x) which are sharp as α → −1. We indicate the applicability of the results to more general… Expand

Quasi-orthogonality with applications to some families of classical orthogonal polynomials

- Mathematics
- 2004

In this paper, we study the quasi-orthogonality of orthogonal polynomials. New results on the location of their zeros are given in two particular cases. Then these results are applied to Gegenbauer,… Expand