# OrthogonalPolynomialFunctions R¶

This package provides orthogonal polynomials as functions on a ring.

chebyshevT: (NonNegativeInteger, R) -> R
chebyshevT(n, x) is the n-th Chebyshev polynomial of the first kind, T[n](x). These are defined by (1-t*x)/(1-2*t*x+t^2) = sum(T[n](x) *t^n, n = 0..).
chebyshevU: (NonNegativeInteger, R) -> R
chebyshevU(n, x) is the n-th Chebyshev polynomial of the second kind, U[n](x). These are defined by 1/(1-2*t*x+t^2) = sum(T[n](x) *t^n, n = 0..).
hermiteH: (NonNegativeInteger, R) -> R
hermiteH(n, x) is the n-th Hermite polynomial, H[n](x). These are defined by exp(2*t*x-t^2) = sum(H[n](x)*t^n/n!, n = 0..).
laguerreL: (NonNegativeInteger, NonNegativeInteger, R) -> R
laguerreL(m, n, x) is the associated Laguerre polynomial, L<m>[n](x). This is the m-th derivative of L[n](x).
laguerreL: (NonNegativeInteger, R) -> R
laguerreL(n, x) is the n-th Laguerre polynomial, L[n](x). These are defined by exp(-t*x/(1-t))/(1-t) = sum(L[n](x)*t^n/n!, n = 0..).
legendreP: (NonNegativeInteger, R) -> R if R has Algebra Fraction Integer
legendreP(n, x) is the n-th Legendre polynomial, P[n](x). These are defined by 1/sqrt(1-2*x*t+t^2) = sum(P[n](x)*t^n, n = 0..).