OrthogonalPoynomials - crowlogic/arb4j GitHub Wiki

Polynomial Family	Weight Function W(x)	Interval
Hermite	$$e^{-x^2}$$	(-∞, ∞)
Laguerre	$$x^{\alpha}e^{-x}$$	[0, ∞)
Jacobi	$$(1-x)^{\alpha}(1+x)^{\beta}$$	[-1, 1]
Chebyshev (1st kind)	$$\frac{1}{\sqrt{1-x^2}}$$	[-1, 1]
Chebyshev (2nd kind)	$$\sqrt{1-x^2}$$	[-1, 1]
Legendre	1	[-1, 1]
Gegenbauer	$$(1-x^2)^{\alpha-\frac{1}{2}}$$	[-1, 1]

These polynomials satisfy the orthogonality relation:

$$\int P_m(x)P_n(x)W(x)dx = 0, \quad m \neq n $$

where the integral ranges over the given interval.