diff --git a/Project.toml b/Project.toml index 762517a..be1bf1b 100644 --- a/Project.toml +++ b/Project.toml @@ -3,6 +3,7 @@ uuid = "b30e2e7b-c4ee-47da-9d5f-2c5c27239acd" authors = ["Sheehan Olver "] version = "0.14.4" + [deps] ArrayLayouts = "4c555306-a7a7-4459-81d9-ec55ddd5c99a" BandedMatrices = "aae01518-5342-5314-be14-df237901396f" diff --git a/docs/src/index.md b/docs/src/index.md index 4267595..9801d28 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -10,12 +10,12 @@ CurrentModule = ClassicalOrthogonalPolynomials We follow the [Digital Library of Mathematical Functions](https://dlmf.nist.gov/18.3), which defines the following classical orthogonal polynomials: -1. Legendre: `P_n(x)` -2. Chebyshev (1st kind, 2nd kind): `T_n(x)`, `U_n(x)` -3. Ultraspherical: `C_n^{(λ)}(x)` -4. Jacobi: `P_n^{(a,b)}(x)` -5. Laguerre: `L_n^{(α)}(x)` -6. Hermite: `H_n(x)` +1. Legendre: $P_n(x)$, defined over $[-1, 1]$ with weight $w(x) = 1$. +2. Chebyshev (1st kind, 2nd kind): $T_n(x)$ and $U_n(x)$, defined over $[-1, 1]$ with weights $w(x) = 1/\sqrt{1-x^2}$ and $w(x) = \sqrt{1-x^2}$, respectively. +3. Ultraspherical: $C_n^{(\lambda)}(x)$, defined over $[-1, 1]$ with weight $w(x) = (1-x^2)^{\lambda-1/2}$. +4. Jacobi: $P_n^{(a,b)}(x)$, defined over $[-1, 1]$ with weight $w(x) = (1-x)^a(1+x)^b$. +5. Laguerre: $L_n^{(\alpha)}(x)$, defined over $[0, ∞)$ with weight $w(x) = x^\alpha \mathrm{e}^{-x}$. +6. Hermite: $H_n(x)$, defined over $(-∞, ∞)$ with weight $w(x) = \mathrm{e}^{-x^2}$. These special polynomials have many applications and can be used as a basis for any function given their domain conditions are met, however these polynomials have some advantages due to their formulation: