Inclusion Theorems for the Zeros of Certain Recursively Generated Polynomials

Abstract

Consider the sequence $\{ \varphi _n \}$ of monic polynomials generated by the recurrence relation $\varphi _0 = 1$, $\varphi _1 = z - b_0 ,$, $\varphi _{n + 1} = (z - b_n )\varphi _n - c_n \varphi _{n - 1} $, $n \geqq 1,b_n ,c_n $ complex and independent of z, $c_n \ne 0$. This paper contains two theorems of the inclusion type for zeros of the polynomials $\varphi _n $. The results are obtained by applying majorization techniques to certain tridiagonal matrices associated with such polynomial sequences. If each $b_n $ is real and each $c_n > 0$, the polynomials are orthogonal with respect to some distribution on some set of points of the real line, and the zeros are real. Two additional theorems deal with the smallest interval containing these zeros, i.e., the true interval of orthogonality.