Determinantal polynomials of a weighted shift matrix with palindromic geometric weights

The numerical radius of a weighted shift operator with geometric weights

Let T be a weighted shift operator T on the Hilbert space 2(N) with geometric weights. Then the numerical range of T is a closed disk about the origin, and its numerical radius is determined in terms of the reciprocal of the minimum positive root of a hypergeometric function. This function is related to two Rogers-Ramanujan identities.

Smith Forms of Palindromic Matrix Polynomials

Many applications give rise to matrix polynomials whose coefficients have a kind of reversal symmetry, a structure we call palindromic. Several properties of scalar palindromic polynomials are derived, and together with properties of compound matrices, used to establish the Smith form of regular and singular T -palindromic matrix polynomials over arbitrary fields. The invariant polynomials are ...

Determinantal representations of hyperbolic forms via weighted shift matrices

Weighted inequalities for generalized polynomials with doubling weights

Many weighted polynomial inequalities, such as the Bernstein, Marcinkiewicz, Schur, Remez, Nikolskii inequalities, with doubling weights were proved by Mastroianni and Totik for the case [Formula: see text], and by Tamás Erdélyi for [Formula: see text]. In this paper we extend such polynomial inequalities to those for generalized trigonometric polynomials. We also prove the large sieve for gene...

A Family of Palindromic Polynomials

The relation between the roots of a polynomial P (z) and its coefficients was one of the driving forces in the development of Algebra. The desire to obtain a closed-form expression for the roots of P (z) = 0 ended with a negative solution at the hands of N. Abel and E. Galois at the beginning of the 19 century: a generic polynomial equation of degree 5 or more cannot be solved by radicals. The ...