Orthogonal polynomials

Summary

A sequence of orthogonal polynomials is a sequence $p_0, p_1, p_2, \dots$ with $\deg p_n = n$ that are pairwise orthogonal. Orthogonality refers to the inner product $\int p q \, d\mu$ of a measure $\mu$ on the real line. They arise from the monomials $1, x, x^2, \dots$ by Gram–Schmidt orthogonalization, and they are unique up to constant factors. The classical families of Hermite, Laguerre, Jacobi, Legendre and Chebyshev polynomials are examples. They satisfy a three-term recurrence, their zeros are real and interlace, and they are used in approximation, numerical integration, spectral theory and random matrices.