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Private Documentation

Legendre.Plm_00 — Function
Plm_00(::N, ::Type{T}) where {N<:AbstractLegendreNorm, T}

Returns the initial condition $P_0^0(x)$ for the associated Legendre recursions based on the normalization choice N for numeric type T.

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Legendre.Plm_α — Function
Plm_α(norm::N, ::Type{T}, l::Integer, m::Integer) where {N<:AbstractLegendreNorm, T}

Returns the coefficient $α_ℓ^m$ for the two-term recursion relation

\[ P_{ℓ+1}^{m}(x) = α_{ℓ+1}^m x P_ℓ^m(x) - β_{ℓ+1}^m P_{ℓ-1}^m(x)\]

where $α_ℓ^m$ is appropriate for the choice of normalization N.

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Legendre.Plm_β — Function
Plm_β(norm::N, ::Type{T}, l::Integer, m::Integer) where {N<:AbstractLegendreNorm, T}

Returns the coefficient $β_ℓ^m$ for the two-term recursion relation

\[ P_{ℓ+1}^{m}(x) = α_{ℓ+1}^m x P_ℓ^m(x) - β_{ℓ+1}^m P_{ℓ-1}^m(x)\]

where $β_ℓ^m$ is appropriate for the choice of normalization N.

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Legendre.Plm_μ — Function
Plm_μ(norm::N, ::Type{T}, l::Integer) where {N<:AbstractLegendreNorm, T}

Returns the coefficient $μ_ℓ$ for the single-term recursion relation

\[ P_{ℓ+1}^{ℓ+1}(x) = -μ_{ℓ+1} \sqrt{1-x^2} P_ℓ^ℓ(x)\]

where $μ_ℓ$ is appropriate for the choice of normalization N.

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Legendre.Plm_ν — Function
Plm_ν(norm::N, ::Type{T}, l::Integer) where {N<:AbstractLegendreNorm, T}

Returns the coefficient $ν_ℓ$ for the single-term recursion relation

\[ P_{ℓ+1}^ℓ(x) = ν_{ℓ+1} x P_ℓ^ℓ(x)\]

where $ν_ℓ$ is appropriate for the choice of normalization N.

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