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

Zeros of quasi-orthogonal ultraspherical polynomials
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 ofExpand
Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomials
TLDR
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
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-orthogonalExpand
Zeros of Jacobi polynomials Pn(α,β)$ P_{n}^{(\alpha ,\beta)} $, − 2 < α, β < − 1
The sequence of Jacobi polynomials {Pn(α,β)}n=0∞$\{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
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
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
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 $$-2<\alphaExpand
Zeros of quasi-orthogonal ultraspherical polynomials
Abstract For each fixed value of λ in the range − 3 / 2 λ − 1 / 2 , we prove interlacing properties for the zeros of polynomials, of consecutive and non-consecutive degree, within the sequence ofExpand
Interlacing of zeros of Gegenbauer polynomials of non-consecutive degree from different sequences
  • K. Driver
  • Mathematics, Computer Science
  • Numerische Mathematik
  • 2012
TLDR
It is proved that Stieltjes interlacing holds between the zeros of the kth derivative of $${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
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 oldExpand
Zeros of ultraspherical polynomials and the Hilbert-Klein formulas
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. 11Expand
Trajectories of the zeros of hypergeometric polynomials F(−n, b; 2b; z) for b < − 1/2
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
TLDR
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
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 polynomialExpand
INEQUALITIES FOR THE SMALLEST ZEROS OF LAGUERRE POLYNOMIALS AND THEIR q-ANALOGUES
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 generalExpand
Quasi-orthogonality with applications to some families of classical orthogonal polynomials
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
...
1
2
3
4
...