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1. N. Hale and A. Townsend. <ahref="http://dx.doi.org/10.1137/130932223">A fast, simple, and stable Chebyshev—Legendre transform using and asymptotic formula</a>, SIAM J. Sci. Comput., 36:A148—A167, 2014.
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2. R. M. Slevinsky. <ahref="http://arxiv.org/abs/1602.02618">On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev—Jacobi transform</a>, arXiv:1602.02618, 2016.
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3. A. Townsend, M. Webb, and S. Olver. <ahref="http://arxiv.org/abs/1604.07486">Fast polynomial transforms based on Toeplitz and Hankel matrices</a>, arXiv:1604.07486, 2016.
For 64-bit floating-point arithmetic, the Lambda function uses the asymptotic series for τ in Appendix B of
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I. Bogaert and B. Michiels and J. Fostier, 𝒪(1) computation of Legendre polynomials and Gauss–Legendre nodes and weights for parallel computing, SIAM J. Sci. Comput., 34:C83–C101, 2012.
[cjt(c, α, β)](FastTransforms.md#method__cjt.1) Computes the Chebyshev expansion coefficients
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[gaunt(m::Int64, n::Int64, μ::Int64, ν::Int64)](FastTransforms.md#method__gaunt.1) Calculates the Gaunt coefficients in 64-bit floating-point arithmetic.
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[gaunt{T}(::Type{T}, m::Int64, n::Int64, μ::Int64, ν::Int64)](FastTransforms.md#method__gaunt.2) Calculates the Gaunt coefficients, defined by:
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[icjt(c, α, β)](FastTransforms.md#method__icjt.1) Computes the Jacobi expansion coefficients with parameters ``α`` and ``β``
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[jjt(c, α, β, γ, δ)](FastTransforms.md#method__jjt.1) Computes the Jacobi expansion coefficients with parameters ``γ`` and ``δ``
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[plan_cjt(c::AbstractArray{T, 1}, α, β)](FastTransforms.md#method__plan_cjt.1) Pre-plan optimized DCT-I and DST-I plans and pre-allocate the necessary
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[plan_icjt(c::AbstractArray{T, 1}, α, β)](FastTransforms.md#method__plan_icjt.1) Pre-plan optimized DCT-I and DST-I plans and pre-allocate the necessary
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---
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## Methods [Internal]
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[chebyshevjacobimoments1{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat)](FastTransforms.md#method__chebyshevjacobimoments1.1) Modified Chebyshev moments of the first kind with respect to the Jacobi weight:
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[chebyshevjacobimoments2{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat)](FastTransforms.md#method__chebyshevjacobimoments2.1) Modified Chebyshev moments of the second kind with respect to the Jacobi weight:
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[clenshawcurtisweights{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat)](FastTransforms.md#method__clenshawcurtisweights.1) Compute weights of the Clenshaw—Curtis quadrature rule with a Jacobi weight.
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[clenshawcurtis{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat)](FastTransforms.md#method__clenshawcurtis.1) Compute nodes and weights of the Clenshaw—Curtis quadrature rule with a Jacobi weight.
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[decrementα!(c::AbstractArray{T, 1}, α, β)](FastTransforms.md#method__decrement945.1) Compute Jacobi expansion coefficients in Pₙ^(α-1,β) given Jacobi expansion coefficients in Pₙ^(α,β) in-place.
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[decrementαβ!(c::AbstractArray{T, 1}, α, β)](FastTransforms.md#method__decrement945946.1) Compute Jacobi expansion coefficients in Pₙ^(α-1,α-1) given Jacobi expansion coefficients in Pₙ^(α,α) in-place.
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[decrementβ!(c::AbstractArray{T, 1}, α, β)](FastTransforms.md#method__decrement946.1) Compute Jacobi expansion coefficients in Pₙ^(α,β-1) given Jacobi expansion coefficients in Pₙ^(α,β) in-place.
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[fejer1{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat)](FastTransforms.md#method__fejer1.1) Compute nodes and weights of Fejer's first quadrature rule with a Jacobi weight.
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[fejer2{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat)](FastTransforms.md#method__fejer2.1) Compute nodes and weights of Fejer's second quadrature rule with a Jacobi weight.
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[fejerweights1{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat)](FastTransforms.md#method__fejerweights1.1) Compute weights of Fejer's first quadrature rule with a Jacobi weight.
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[fejerweights2{T<:AbstractFloat}(N::Int64, α::T<:AbstractFloat, β::T<:AbstractFloat)](FastTransforms.md#method__fejerweights2.1) Compute weights of Fejer's second quadrature rule with a Jacobi weight.
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[half(x::Number)](FastTransforms.md#method__half.1) Compute a typed 0.5.
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[incrementα!(c::AbstractArray{T, 1}, α, β)](FastTransforms.md#method__increment945.1) Compute Jacobi expansion coefficients in Pₙ^(α+1,β) given Jacobi expansion coefficients in Pₙ^(α,β) in-place.
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[incrementαβ!(c::AbstractArray{T, 1}, α, β)](FastTransforms.md#method__increment945946.1) Compute Jacobi expansion coefficients in Pₙ^(α+1,α+1) given Jacobi expansion coefficients in Pₙ^(α,α) in-place.
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[incrementβ!(c::AbstractArray{T, 1}, α, β)](FastTransforms.md#method__increment946.1) Compute Jacobi expansion coefficients in Pₙ^(α,β+1) given Jacobi expansion coefficients in Pₙ^(α,β) in-place.
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[pochhammer(x::Number, n::Integer)](FastTransforms.md#method__pochhammer.1) Pochhammer symbol (x)_n = Γ(x+n)/Γ(x) for the rising factorial.
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[stirlingseries(z)](FastTransforms.md#method__stirlingseries.1) Stirling series for Γ(z).
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[two(x::Number)](FastTransforms.md#method__two.1) Compute a typed 2.
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[Λ(x::Float64)](FastTransforms.md#method__923.1) For 64-bit floating-point arithmetic, the Lambda function uses the asymptotic series for τ in Appendix B of
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[Λ(z::Number)](FastTransforms.md#method__923.2) The Lambda function Λ(z) = Γ(z+½)/Γ(z+1) for the ratio of gamma functions.
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[Λ(z::Number, λ₁::Number, λ₂::Number)](FastTransforms.md#method__923.3) The Lambda function Λ(z,λ₁,λ₂) = Γ(z+λ₁)/Γ(z+λ₂) for the ratio of gamma functions.
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[δ(k::Integer, j::Integer)](FastTransforms.md#method__948.1) The Kronecker δ function.
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---
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## Types [Internal]
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[FastTransforms.ToeplitzHankelPlan{S}](FastTransforms.md#type__toeplitzhankelplan.1) Store a diagonally-scaled Toeplitz∘Hankel matrix:
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1. N. Hale and A. Townsend. <ahref="http://dx.doi.org/10.1137/130932223">A fast, simple, and stable Chebyshev—Legendre transform using and asymptotic formula</a>, SIAM J. Sci. Comput., 36:A148—A167, 2014.
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2. R. M. Slevinsky. <ahref="http://arxiv.org/abs/1602.02618">On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev—Jacobi transform</a>, arXiv:1602.02618, 2016.
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3. A. Townsend, M. Webb, and S. Olver. <ahref="http://arxiv.org/abs/1604.07486">Fast polynomial transforms based on Toeplitz and Hankel matrices</a>, arXiv:1604.07486, 2016.
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