`scipy.special._orthogonal:sh_legendre`

source: build-install/usr/lib/python3.14/site-packages/scipy/special/_orthogonal.py:3587

Signature

def sh_legendre(n, monic = False)

Summary

Shifted Legendre polynomial.

Extended Summary

Defined as $P_{n}^{*} (x) = P_{n} (2 x - 1)$ for $P_{n}$ the nth Legendre polynomial.

Parameters

n : int: Degree of the polynomial.
monic : bool, optional: If True, scale the leading coefficient to be 1. Default is False.

Returns

P : orthopoly1d: Shifted Legendre polynomial.

Notes

The polynomials $P_{n}^{*}$ are orthogonal over $[0, 1]$ with weight function 1.

Examples

The shifted Legendre polynomials :math:`P_n^*` are related to the non-shifted polynomials :math:`P_n` by :math:`P_n^*(x) = P_n(2x - 1)`. We can verify this on the interval :math:`[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 :math:`P_n^*` satisfy a recurrence relation obtained by the change of variables :math:`t = 2x - 1` in the standard Legendre recurrence: .. math:: (n+1) P_{n+1}^*(x) = (2n+1)(2x-1)\,P_n^*(x) - n\,P_{n-1}^*(x). This can be easily checked on :math:`[0, 1]` for :math:`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 :math:`[0,1]` with weight 1 can be checked numerically; for example, :math:`P_2^*` is orthogonal to :math:`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)

✗

np.True_

Aliases

scipy.special.sh_legendre