For further details, see Legendre polynomials. Associated Legendre polynomials The Associated Legendre polynomials, denoted where and are integers with , are defined as The m in parentheses (to avoid confusion with an exponent) is a parameter. The m in brackets denotes the m-th derivative of the Legendre polynomial. These "polynomials" are misnamed—they are not polynomials when m is odd. They have a recurrence relation: For fixed m, the sequence are orthogonal over [−1, 1], with weight 1. For given m, are the solutions of Chebyshev polynomials The differential equation is This is Chebyshev's equation. The recurrence relation is Rodrigues' formula is These polynomials have the property that, in the interval of orthogonality, (To prove it, use the recurrence formula.) This means that all their local minima and maxima have values of −1 and +1, that is, the polynomials are "level". Because of this, expansion of functions in terms of Chebyshev polynomials is sometimes used for polynomial approximations in computer math libraries. Some authors use versions of these polynomials that have been shifted so that the interval of orthogonality is [0, 1] or [−2, 2]. There are also Chebyshev polynomials of the second kind, denoted We have: For further details, including the expressions for the first few polynomials, see Chebyshev polynomials. Laguerre polynomials The most general Laguerre-like polynomials, after the domain has been shifted and scaled, are the Associated Laguerre polynomials (also called generalized Laguerre polynomials), denoted . There is a parameter , which can be any real number strictly greater than −1. The parameter is put in parentheses to avoid confusion with an exponent. The plain Laguerre polynomials are simply the version of these: The differential equation is This is Laguerre's equation. The second form of the differential equation is The recurrence relation is Rodrigues' formula is The parameter is closely related to the derivatives of : or, more generally: Laguerre's equation can be manipulated into a form that is more useful in applications: is a solution of This can be further manipulated. When is an integer, and : is a solution of The solution is often expressed in terms of derivatives instead of associated Laguerre polynomials: This equation arises in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. Physicists often use a definition for the Laguerre polynomials that is larger, by a factor of , than the definition used here.