For further details, including the expressions for the first few polynomials, see Laguerre polynomials. Hermite polynomials The differential equation is This is Hermite's equation. The second form of the differential equation is The third form is The recurrence relation is Rodrigues' formula is The first few Hermite polynomials are One can define the associated Hermite functions Because the multiplier is proportional to the square root of the weight function, these functions are orthogonal over with no weight function. The third form of the differential equation above, for the associated Hermite functions, is The associated Hermite functions arise in many areas of mathematics and physics. In quantum mechanics, they are the solutions of Schrödinger's equation for the harmonic oscillator. They are also eigenfunctions (with eigenvalue (−i n) of the continuous Fourier transform. Many authors, particularly probabilists, use an alternate definition of the Hermite polynomials, with a weight function of instead of . If the notation He is used for these Hermite polynomials, and H for those above, then these may be characterized by For further details, see Hermite polynomials. Characterizations of classical orthogonal polynomials There are several conditions that single out the classical orthogonal polynomials from the others. The first condition was found by Sonine (and later by Hahn), who showed that (up to linear changes of variable) the classical orthogonal polynomials are the only ones such that their derivatives are also orthogonal polynomials. Bochner characterized classical orthogonal polynomials in terms of their recurrence relations. Tricomi characterized classical orthogonal polynomials as those that have a certain analogue of the Rodrigues formula. Table of classical orthogonal polynomials The following table summarises the properties of the classical orthogonal polynomials. See also Appell sequence Askey scheme of hypergeometric orthogonal polynomials Polynomial sequences of binomial type Biorthogonal polynomials Generalized Fourier series Secondary measure Sheffer sequence Umbral calculus Notes References Articles containing proofs Orthogonal polynomials Special hypergeometric functions zh:正交多項式