OrthogonalPolynomialFunctions(R)

special.spad line 120 [edit on github]

• R : CommutativeRing

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, 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..).

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).

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..).