Matrix methods in the theory of multiplier sequences

Church, Amber
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University of Guelph

Let gn[infinity]n=0 be a sequence of real or complex numbers and let [gamma] be the linear operator on the space of polynomials which maps 'zn' to [gamma]'nzn' for all ' n'. We study how this operator affects the roots of polynomials. A multiplier sequence of the first kind maps polynomials with all real roots to polynomials of the same class, while a multiplier sequence of the second kind takes a polynomial with all positive roots to a polynomial with all real roots. Multiplier sequences have been studied by such mathematicians as Laguerre, Polya, Schur and Szego, We explore some of their basic results on multiplier sequences. We apply matrix theoretical techniques to give new proofs of these classical results and to provide examples of sequences of the first and second kind. One of the advantages of using matrix techniques in the study of polynomials is that they allow us to use the majorization order to study how dispersed the roots of a polynomial are. We prove two new majorization results for a certain class of multiplier sequences.

linear operator, polynomials, roots of polynomials, matrix theoretical techniques, multiplier sequences