Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices

摘要

We consider a sequence ¦pn¦ of orthogonal polynomials defined by a three-term recurrence formula. Representations and bounds are derived for the endpoints of the smallest interval containing the (real and distinct) zeros of Pn in terms of the parameters in the recurrence relation. These results are brought to light by viewing (−1)nPn as the characteristic polynomial of a sign-symmetric tri-diagonal matrix of order n. Our findings are subsequently used to obtain new proofs for a number of bounds on the endpoints of the true and limit intervals of orthogonality for the sequence ¦pn¦.