Moments and orthogonal polynomials

Summary

The orthogonal polynomials of a measure on the real line depend only on its moments. Given a moment sequence $(m_n)$ whose Hankel matrices are positive definite, the monic orthogonal polynomials are explicit quotients of determinants built from the $m_n$. This link lets one pass between a sequence of numbers, a family of polynomials, a three-term recurrence and a measure. It is a main tool in the solution of the Hamburger moment problem and in the theory of Gaussian quadrature.