In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative "weight function" w precisely if

In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as

then the orthogonal polynomials are simply orthogonal vectors in this inner product space.

A polynomial sequence p_n(x) for n = 0, 1, 2, ... , where p_n(x) has degree n, is said to be a sequence of orthogonal polynomials with respect to a "weight function" w when any two of them are orthogonal with respect to that weight function, i.e.,

For example:

• The Hermite polynomials are orthogonal with respect to a normal probability distribution.

• The Chebyshev polynomials are orthogonal with respect to the weight function

• The Legendre polynomials are orthogonal with respect to the uniform probability distribution on the interval [−1, 1].

See also generalized Fourier series.