Interlacing of the zeros of Jacobi polynomials with different parameters

Abstract

We prove results for the interlacing of zeros of Jacobi polynomials of the same or adjacent degree as one or both of the parameters are shifted continuously within a certain range. Numerical examples are given to illustrate situations where interlacing fails to occur.

On the Interlacing of Zeros of Linear Combinations of Jacobi Polynomials from Different Sequences

“…It was shown in [7,Theorem 2.6], that the zeros of P (α,β) n interlace with the zeros of P (α−k,β+t) n for t, k ∈ (0, 2]. The result then follows from Lemma 2.1(a).…”

“…[7,Theorem 2.2]). We deduce that there is at least one zero of E (α,β,0,1) n between any two consecutive zeros of P (α−1,β+1) n+1 and the result follows.…”

“…Indeed, even in the simplest case when t = k = 1 and n = 6, α = 2.3, β = 3.2, ν = 3, the zeros of E are given by the larger green dots, while those of P Lemma 2.1(a) requires that the zeros of p n and q n are interlacing. We showed in [7] that the zeros of Jacobi polynomials of the same degree do not interlace when both the parameters α and β are increased simultaneously. Using this, it is not difficult to construct examples with p n = P (α,β) n and q n = P (α+k,β+t) n where the zeros of p n + νq n and p n or q n do not interlace.…”

“…Our method of proof makes extensive use of the interlacing property of the zeros of P (α,β) n and P (α ′ ,β ′ ) m for m = n and m = n − 1 and (α ′ , β ′ ) = (α ± t, β ± k), 0 < t ≤ 2 and 0 < k ≤ 2, proved in [7]. These results proved a conjecture posed by R. Askey in 1989 (cf.…”

Monotonicity of zeros of polynomials orthogonal with respect to an even weight function

“…The manner in which the zeros of a polynomial change as the parameter changes can be used to study comparison and interlacing properties of the zeros [17][18][19][20][21]. Markoff's theorem can be used to show that the zeros of classical orthogonal polynomials like Laguerre and Jacobi polynomials are monotone functions of the parameter(s) involved by using the derivative of the weight function with respect to the parameter(s).…”