In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials and the Legendre polynomials as special cases. These are frequently given by the Rodrigues' formula. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was pursued by A. A. Markov and T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (quadrature rules), probability theory, representation theory (of Lie groups, quantum groups, and related objects), enumerative combinatorics, algebraic combinatorics, mathematical physics (the theory of random matrices, integrable systems, etc.), and number theory. Some of the mathematicians who have worked on orthogonal polynomials include Gábor Szegő, Sergei Bernstein, Naum Akhiezer, Arthur Erdélyi, Yakov Geronimus, Wolfgang Hahn, Theodore Seio Chihara, Mourad Ismail, Waleed Al-Salam, Richard Askey, and Rehuel Lobatto.
Definition for 1-variable case for a real measure Given any non-decreasing function α on the real numbers, we can define the Lebesgue–Stieltjes integral
∫ f ( x ) d α ( x ) {\displaystyle \int f(x)\,d\alpha (x)}
of a function f. If this integral is finite for all polynomials f, we can define an inner product on pairs of polynomials f and g by
⟨ f , g ⟩ = ∫ f ( x ) g ( x ) d α ( x ) . {\displaystyle \langle f,g\rangle =\int f(x)g(x)\,d\alpha (x).}
This operation is a positive semidefinite inner product on the vector space of all polynomials, and is positive definite if the function α has an infinite number of points of growth. It induces a notion of orthogonality in the usual way, namely that two polynomials are orthogonal if their inner product is zero. Then the sequence (Pn)∞n=0 of orthogonal polynomials is defined by the relations
deg P n = n , ⟨ P m , P n ⟩ = 0 for m ≠ n . {\displaystyle \deg P_{n}=n~,\quad \langle P_{m},\,P_{n}\rangle =0\quad {\text{for}}\quad m\neq n~.}
In other words, the sequence is obtained from the sequence of monomials 1, x, x2, … by the Gram–Schmidt process with respect to this inner product. Usually the sequence is required to be orthonormal, namely,
⟨ P n , P n ⟩ = 1 , {\displaystyle \langle P_{n},P_{n}\rangle =1,}
however, other normalisations are sometimes used.
Absolutely continuous case Sometimes we have
d α ( x ) = W ( x ) d x {\displaystyle d\alpha (x)=W(x)\,dx}
where
W : [ x 1 , x 2 ] → R {\displaystyle W:[x_{1},x_{2}]\to \mathbb {R} }
is a non-negative function with support on some interval [x1, x2] in the real line (where x1 = −∞ and x2 = ∞ are allowed). Such a W is called a weight function. Then the inner product is given by
⟨ f , g ⟩ = ∫ x 1 x 2 f ( x ) g ( x ) W ( x ) d x . {\displaystyle \langle f,g\rangle =\int _{x_{1}}^{x_{2}}f(x)g(x)W(x)\,dx.}
However, there are many examples of orthogonal polynomials where the measure dα(x) has points with non-zero measure where the function α is discontinuous, so cannot be given by a weight function W as above.
Examples of orthogonal polynomials The most commonly used orthogonal polynomials are orthogonal for a measure with support in a real interval. This includes:
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