orthax.recurrence.GeneralizedLaguerre

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

Recurrence relation for Generalized Laguerre polynomials \(L^\alpha_n(x)\)

Generalized Laguerre polynomials are orthogonal on the interval (0, inf) with the weight function \(w(x) = x^\alpha e^{-x}\)

Parameters:

  • alpha (float > -1) – Hyperparameter α.

  • 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__(alpha: Array | ndarray | bool | number | bool | int | float | complex, scale: str = 'standard')Source

Methods

__init__(alpha[, 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.

alpha