scipy.special.

sh_legendre#

scipy.special.sh_legendre(n, monic=False)[source]#

Shifted Legendre polynomial.

Defined as \(P^*_n(x) = P_n(2x - 1)\) for \(P_n\) the nth Legendre polynomial.

Parameters:

nint: Degree of the polynomial.
monicbool, optional: If True, scale the leading coefficient to be 1. Default is False.

Returns:

Porthopoly1d: Shifted Legendre polynomial.

See also

scipy.special.legendre
scipy.special.roots_sh_legendre

Notes

The polynomials \(P^*_n\) are orthogonal over \([0, 1]\) with weight function 1.

Examples

The shifted Legendre polynomials \(P_n^*\) are related to the non-shifted polynomials \(P_n\) by \(P_n^*(x) = P_n(2x - 1)\). We can verify this on the interval \([0, 1]\):

>>> import numpy as np
>>> from scipy.special import sh_legendre, legendre
>>> from scipy.integrate import trapezoid
>>> x = np.arange(0.0, 1.0, 0.01)
>>> n = 3
>>> np.allclose(sh_legendre(n)(x), legendre(n)(2*x - 1))
True

The polynomials \(P_n^*\) satisfy a recurrence relation obtained by the change of variables \(t = 2x - 1\) in the standard Legendre recurrence:

\[(n+1) P_{n+1}^*(x) = (2n+1)(2x-1)\,P_n^*(x) - n\,P_{n-1}^*(x).\]

This can be easily checked on \([0, 1]\) for \(n = 3\):

>>> n = 3
>>> x = np.linspace(0.0, 1.0, 101)
>>> lhs = (n + 1) * sh_legendre(n + 1)(x)
>>> rhs = (
...     (2*n + 1) * (2*x - 1) * sh_legendre(n)(x)
...     - n * sh_legendre(n - 1)(x)
... )
>>> np.allclose(lhs, rhs)
True

Orthogonality over \([0,1]\) with weight 1 can be checked numerically; for example, \(P_2^*\) is orthogonal to \(P_3^*\):

>>> x = np.linspace(0.0, 1.0, 400)
>>> y = sh_legendre(2)(x) * sh_legendre(3)(x)
>>> np.isclose(trapezoid(y, x), 0.0, atol=1e-12)
True