Newton sum rules of zeros of semiclassical orthogonal polynomials

摘要

The distribution of zeros of the semiclassical orthogonal polynomials with weights w̄(x) = π(x)wc(x), wc(x) denoting a classical weight function and π(x) being equal to |x − c| or x2 + c2, is investigated via the Newton sum rules of zeros (i.e, the sums of the rth powers of zeros). Recursion relations satisfied by these sum rules for semiclassical Legendre, Laguerre and Hermite polynomials are explicitly given. Extensions to other semiclassical polynomials are indicated.