Circular interlacing with reciprocal polynomials

Self-interlacing polynomials II: Matrices with self-interlacing spectrum

An n×n matrix is said to have a self-interlacing spectrum if its eigenvalues λk , k = 1, . . . ,n, are distributed as follows: λ1 >−λ2 > λ3 > · · ·> (−1)λn > 0. A method for constructing sign definite matrices with self-interlacing spectrum from totally nonnegative ones is presented. This method is applied to bidiagonal and tridiagonal matrices. In particular, a result by O. Holtz on the spectr...

Polynomials , roots , and interlacing

6.1 Interlacing Families of Polynomials

In this lecture we introduce the concept of interlacing for polynomials. This concept provide a convenient way of reasoning about orderings of the roots of real-rooted polynomials. Specifically, it provides a way of tying the roots of the average polynomial (obtained by averaging the coefficients) to the roots of the individual polynomials. This can for example —in the spirit of the probabilist...

Interlacing and asymptotic properties of Stieltjes polynomials

Polynomial solutions to the generalized Lamé equation, the Stieltjes polynomials, and the associated Van Vleck polynomials have been studied since the 1830’s, beginning with Lamé in his studies of the Laplace equation on an ellipsoid, and in an ever widening variety of applications since. In this paper we show how the zeros of Stieltjes polynomials are distributed and present two new interlacin...

Reciprocal cyclotomic polynomials

Let Ψn(x) be the monic polynomial having precisely all non-primitive nth roots of unity as its simple zeros. One has Ψn(x) = (x n − 1)/Φn(x), with Φn(x) the nth cyclotomic polynomial. The coefficients of Ψn(x) are integers that like the coefficients of Φn(x) tend to be surprisingly small in absolute value, e.g. for n < 561 all coefficients of Ψn(x) are ≤ 1 in absolute value. We establish variou...