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Add intervals/weights to the definitions in the docs #222

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Merged
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Jan 28, 2025
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Add intervals/weights to the definitions in the docs #222

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08402f0
Expand on the definitions
DanielVandH Dec 8, 2024
c38a99b
Merge branch 'main' into dv/docdef
dlfivefifty Jan 28, 2025
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1 change: 1 addition & 0 deletions Project.toml
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Expand Up @@ -3,6 +3,7 @@ uuid = "b30e2e7b-c4ee-47da-9d5f-2c5c27239acd"
authors = ["Sheehan Olver <[email protected]>"]
version = "0.14.4"


[deps]
ArrayLayouts = "4c555306-a7a7-4459-81d9-ec55ddd5c99a"
BandedMatrices = "aae01518-5342-5314-be14-df237901396f"
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12 changes: 6 additions & 6 deletions docs/src/index.md
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Expand Up @@ -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:

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