Interlacing Of Zeros Research Articles

Interlacing of zeros of quasi-orthogonal Meixner polynomials

We consider the interlacing of zeros of polynomials within the sequences of quasi-orthogonal order one Meixner polynomials characterised by −β, c ∈ (0, 1). The interlacing of zeros of quasi-orthogonal Meixner polynomials Mn(x; β, c) with the zeros of their nearest orthogonal counterparts Ml(x; β + k, c), l, n ∈ ℕ, k ∈ {1, 2}, is also discussed.

Non-definite Sturm-Liouville problems for the p-Laplacian

For a weighted Sturm-Liouville-type problem of the form −Δpy = (p−1)(λr−q)sgny|y|p−1 , on (0,1) with Sturmian-type boundary conditions (Δp being the p -Laplacian), we examine the structure, asymptotics and parametric dependence of the eigenvalues, together with properties of the eigenfunctions such as oscillation and interlacing of zeros. We discuss definitions and consequences of left and right (semi-) definiteness, and also the fully indefinite case. Mathematics subject classification (2010): 34B25, 46D05, 47E05.

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

We investigate the interlacing of the zeros of linear combinations pn +aqm with the zeros of the components pn and qm , where {pn }∞ n=0 and {q m }∞ m=0 are different sequences of Jacobi polynomials. The results we prove hold when pn and qm are Jacobi polynomials P n (α,β)(x) and P m (α′,β′)(x) for certain values of α′ and β′ with m = n or m = n – 1. Numerical counterexamples are given in situations where interlacing fails to occur. We also show that the zeros of the linear combination p n + aq m interlace with the zeros of some Jacobi polynomials besides the components of the linear combination.

Conjecture on the interlacing of zeros in complex Sturm–Liouville problems

The zeros of the eigenfunctions of self-adjoint Sturm–Liouville eigenvalue problems interlace. For these problems interlacing is crucial for completeness. For the complex Sturm–Liouville problem associated with the Schrödinger equation for a non-Hermitian 𝒫𝒯-symmetric Hamiltonian, completeness and interlacing of zeros have never been examined. This paper reports a numerical study of the Sturm–Liouville problems for three complex potentials, the large-N limit of a −(ix)N potential, a quasiexactly-solvable −x4 potential, and an ix3 potential. In all cases the complex zeros of the eigenfunctions exhibit a similar pattern of interlacing and it is conjectured that this pattern is universal. Understanding this pattern could provide insight into whether the eigenfunctions of complex Sturm–Liouville problems form a complete set.