Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials

Driver, Simon P.; Jordaan, Kerstin

doi:10.1007/s00211-007-0100-3

Cited by 25 publications

(24 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [4,Theorem 3.4], it is shown that the zeros of C λ n and C λ+t n are interlacing for 0 < t ≤ 2 while values of n and λ can easily be chosen such that interlacing breaks down for the zeros of C λ n and C λ+3 n . We have seen in this paper, however, that the parameters λ and λ + t can vary by more than two (provided the degree of the polynomial is suitably modified) and Stieltjes interlacing of zeros will still occur.…”

Section: Remarksmentioning

confidence: 99%

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

Driver

2011

Numer. Math.

Self Cite

View full text Add to dashboard Cite

A theorem due to Stieltjes' states that if { p n } ∞ n=0 is any orthogonal sequence then, between any two consecutive zeros of p k , there is at least one zero of p n whenever k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends to the zeros of Gegenbauer polynomials C λ n+1 and C λ+t n−1 , λ > − 1 2 , if 0 < t ≤ k + 1, and also to the zeros of C λ n+1 and C λ+k n−2 if k ∈ {1, 2, 3}. More generally, we prove that Stieltjes interlacing holds between the zeros of the kth derivative of C λ n and the zeros of C λ n+1 , k ∈ {1, 2, . . . , n − 1} and we derive associated polynomials that play an analogous role to the de Boor-Saff polynomials in completing the interlacing process of the zeros.

show abstract

Section: Remarksmentioning

confidence: 99%

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

Driver

2011

Numer. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…2 In this paper, we will be more interested in obtaining the new interlacing properties that lie behind the continuity of these coefficients, relating them to properties of classical orthogonal polynomials and Bessel functions. In particular, we will revisit the recent results of Driver and Jordaan [10] (see also [9]) regarding the interlacing properties of zeros of Gegenbauer and Laguerre classical polynomials with shifted parameters. In their paper, they make use of a theorem due to Markov, which gives information regarding the ordering of the zeros of orthogonal polynomials associated with different weights.…”

Section: Introductionmentioning

confidence: 97%

Interlacing of the zeros of contiguous hypergeometric functions

Segura

2008

Numer Algor

View full text Add to dashboard Cite

It is well-known that hypergeometric functions satisfy first order difference-differential equations (DDEs) with rational coefficients, relating the first derivative of hypergeometric functions with functions of contiguous parameters (with parameters differing by integer numbers). However, maybe it is not so well known that the continuity of the coefficients of these DDEs implies that the real zeros of such contiguous functions are interlaced. Using this property, we explore interlacing properties of hypergeometric and confluent hypergeometric functions (Bessel functions and Hermite, Laguerre and Jacobi polynomials as particular cases).

show abstract

“…The interlacing property cannot hold for larger integer shift of the parameter, since the q-ultraspherical polynomials tend to the Gegenbauer polynomials when q → 1 and for the latter the interlacing with a shift of 3 fails (c.f. [8] …”

Section: Acta Mathematica Hungarica 2009mentioning

confidence: 99%

“…This result matches well with the one for the interlacing property of the zeros of the (common, not q-) Laguerre polynomials (cf. [8]). The Laguerre polynomials have the interlacing property for a fixed degree up to a shift of parameter value by 2, but the interlacing property in general fails at a shift by 3.…”

Section: Interlacing Of the Zeros Of The Q-laguerre Polynomials With mentioning

confidence: 99%

See 1 more Smart Citation

Mixed recurrence relations and interlacing of the zeros of some q-orthogonal polynomials from different sequences

Jordaan

Toókos

2010

Acta Math Hung

View full text Add to dashboard Cite

Abstract. We use the generating functions of some q-orthogonal polynomials to obtain mixed recurrence relations involving polynomials with shifted parameter values. These relations are used to prove interlacing results for the zeros of AlSalam-Chihara, continuous q-ultraspherical, q-Meixner-Pollaczek and q-Laguerre polynomials of the same or adjacent degree as one of the parameters is shifted by integer values or continuously within a certain range. Numerical examples are given to illustrate situations where the zeros do not interlace.

show abstract

Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials

Cited by 25 publications

References 16 publications

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

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

Interlacing of the zeros of contiguous hypergeometric functions

Mixed recurrence relations and interlacing of the zeros of some q-orthogonal polynomials from different sequences

Contact Info

Product

Resources

About