# OrthogonalPolynomialFunctions RΒΆ

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..)`.