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README.md

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This package implements classical orthogonal polynomials.

docs/src/index.md

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We follow the [Digital Library of Mathematical Functions](https://dlmf.nist.gov/18.3),
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which defines the following classical orthogonal polynomials:
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1. Legendre: $P_n(x)$
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2. Chebyshev (1st kind, 2nd kind): $T_n(x)$, $U_n(x)$
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3. Ultraspherical: $C_n^{(λ)}(x)$
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4. Jacobi: $P_n^{(a,b)}(x)$
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5. Laguerre: $L_n^{(α)}(x)$
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6. Hermite: $H_n(x)$
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1. Legendre: ``P_n(x)``
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2. Chebyshev (1st kind, 2nd kind): ``T_n(x)``, ``U_n(x)``
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3. Ultraspherical: ``C_n^{(λ)}(x)``
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4. Jacobi: ``P_n^{(a,b)}(x)``
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5. Laguerre: ``L_n^{(α)}(x)``
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6. Hermite: ``H_n(x)``
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## Evaluation
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## Expansions
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We view a function expansion in say Chebyshev polynomials in terms of continuum arrays as follows:
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$$
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```math
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f(x) = \sum_{k=0}^∞ c_k T_k(x) = \begin{bmatrix}T_0(x) | T_1(x) | … \end{bmatrix}
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\begin{bmatrix}c_0\\ c_1 \\ \vdots \end{bmatrix} = T[x,:] * 𝐜
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$$
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```
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To be more precise, we think of functions as continuum-vectors. Here is a simple example:
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```jldoctest
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julia> f = T * [1; 2; 3; zeros(∞)]; # T_0(x) + T_1(x) + T_2(x)
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julia> f[0.1]
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-1.74
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```
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To find the coefficients for a given function we consider this as the problem of finding $𝐜$
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such that $T*𝐜 == f$, that is:
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To find the coefficients for a given function we consider this as the problem of finding ``𝐜``
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such that ``T*𝐜 == f``, that is:
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```julia
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julia> T \ f
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vcat(3-element Vector{Float64}, ℵ₀-element FillArrays.Zeros{Float64, 1, Tuple{InfiniteArrays.OneToInf{Int64}}} with indices OneToInf()) with indices OneToInf():
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```
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For a function given only pointwise we broadcast over `x`, e.g., the following are the coefficients of $\exp(x)$:
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For a function given only pointwise we broadcast over `x`, e.g., the following are the coefficients of ``\exp(x)``:
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```julia
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julia> x = axes(T, 1);
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## Jacobi matrices
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Orthogonal polynomials satisfy well-known three-term recurrences:
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$$
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```math
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x p_n(x) = c_{n-1} p_{n-1}(x) + a_n p_n(x) + b_n p_{n+1}(x).
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$$
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```
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In continuum-array language this has the form of a comuting relationship:
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$$
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```math
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x \begin{bmatrix} p_0 | p_1 | \cdots \end{bmatrix} = \begin{bmatrix} p_0 | p_1 | \cdots \end{bmatrix} \begin{bmatrix} a_0 & c_0 \\ b_0 & a_1 & c_1 \\ & b_1 & a_2 & \ddots \\ &&\ddots & \ddots \end{bmatrix}
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$$
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```
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We can therefore find the Jacobi matrix naturally as follows:
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```julia
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julia> T \ (x .* T)

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