orthax.recurrence.Gegenbauer

class orthax.recurrence.Gegenbauer(lmbda: Array | ndarray | bool | number | bool | int | float | complex, scale: str = 'standard')Source

Recurrence relation for Gegenbauer polynomials \(C^\lambda_n(x)\)

Also known as Ultraspherical harmonics.

Gegenbauer polynomials are orthogonal on the interval (-1, 1) with the weight function \(w(x) = (1-x^2)^{\lambda - 1/2}\)

Parameters:

• lmbda (float > -1/2, != 0) – Hyperparameter λ.

• scale ({"standard", "monic", "normalized"}) – “standard” corresponds to the common scaling found in textbooks such as Abramowitz & Stegun. “monic” scales them such that the leading coefficient is 1. “normalized” scales them to have a weighted norm of 1.

__init__(lmbda: Array | ndarray | bool | number | bool | int | float | complex, scale: str = 'standard')Source

Methods

__init__(lmbda[, scale])
a(k)	a coefficients of the monic three term recurrence relation.
b(k)	b coefficients of the monic three term recurrence relation.
g(k)	Weighted norm of the kth monic orthogonal polynomial.
m(k)	Coefficient of x**k in the kth polynomial in the desired normalization.
weight(x)	Weight function defining inner product.

Attributes

domain	Lower and upper bounds for inner product defining orthogonality.
lmbda