ORTHOGONAL POLYNOMIAL

any of an infinite collection of special polynomials Pn(x), with the subscript n indicating the degree of the polynomial. Their importance derives from their being solutions of differential equations and from the possibility of representing any (continuous) function as a (possibly infinite) sum of these polynomials, a device often useful in the differential equations of physics and engineering. The simplest of these polynomials are the Legendre polynomials. The Legendre polynomials can be obtained by calculating the nth derivative of (x2 - 1)n/2nn!. The polynomial Pn(x) satisfies the so-called second-order Legendre differential equation. They are orthogonal on the interval from -1 to +1, meaning that the integral of the product of any two such functions from -1 to +1 is zero. There are also other classes of orthogonal polynomials, such as Chebyshev and Hermite polynomials, which involve intervals other than -1 to +1.