In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials).

They have many important applications in such areas as mathematical physics (in particular, the theory of random matrices), approximation theory, numerical analysis, and many others.

Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions to solve the moment problem by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials.

For given polynomials ${\displaystyle Q,L:\mathbb {R} \to \mathbb {R} }$ and ${\displaystyle \forall \,n\in \mathbb {N} _{0}}$ the classical orthogonal polynomials ${\displaystyle f_{n}:\mathbb {R} \to \mathbb {R} }$ are characterized by being solutions of the differential equation

with to be determined constants ${\displaystyle \lambda _{n}\in \mathbb {R} }$. The Wikipedia article Rodrigues' formula has a proof that the polynomials obtained from the Rodrigues' formula obey a differential equation of this form and also derives ${\displaystyle \lambda _{n}}$.

There are several more general definitions of orthogonal classical polynomials; for example, Andrews & Askey (1985) use the term for all polynomials in the Askey scheme.

In general, the orthogonal polynomials ${\displaystyle P_{n}}$ with respect to a weight ${\displaystyle W:\mathbb {R} \rightarrow \mathbb {R} ^{+}}$ satisfy

The relations above define ${\displaystyle P_{n}}$ up to multiplication by a number. Various normalisations are used to fix the constant, e.g.