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add the generic ratio of gamma functions Λ(z,λ₁,λ₂) = Γ(z+λ₁)/Γ(z+λ₂)
add jac2jac for moving one set of parameters
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‎README.md

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1. N. Hale and A. Townsend. <a href="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. <a href="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. <a href="http://arxiv.org/abs/1604.07486">Fast polynomial transforms based on Toeplitz and Hankel matrices</a>, arXiv:1604.07486, 2016.

‎docs/api/FastTransforms.md

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*source:*
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[FastTransforms/src/cjt.jl:127](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/cjt.jl#L127)
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[FastTransforms/src/cjt.jl:127](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/cjt.jl#L127)
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*source:*
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[FastTransforms/src/gaunt.jl:24](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/gaunt.jl#L24)
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[FastTransforms/src/gaunt.jl:24](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/gaunt.jl#L24)
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*source:*
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[FastTransforms/src/gaunt.jl:14](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/gaunt.jl#L14)
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[FastTransforms/src/gaunt.jl:14](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/gaunt.jl#L14)
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*source:*
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[FastTransforms/src/cjt.jl:135](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/cjt.jl#L135)
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[FastTransforms/src/cjt.jl:135](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/cjt.jl#L135)
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*source:*
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[FastTransforms/src/cjt.jl:143](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/cjt.jl#L143)
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[FastTransforms/src/cjt.jl:143](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/cjt.jl#L143)
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[FastTransforms/src/cjt.jl:157](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/cjt.jl#L157)
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[FastTransforms/src/cjt.jl:157](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/cjt.jl#L157)
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[FastTransforms/src/cjt.jl:176](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/cjt.jl#L176)
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[FastTransforms/src/cjt.jl:176](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/cjt.jl#L176)
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## Internal
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[FastTransforms/src/specialfunctions.jl:385](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L385)
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[FastTransforms/src/specialfunctions.jl:399](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L399)
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[FastTransforms/src/specialfunctions.jl:403](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L403)
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[FastTransforms/src/specialfunctions.jl:417](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L417)
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[FastTransforms/src/clenshawcurtis.jl:12](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/clenshawcurtis.jl#L12)
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[FastTransforms/src/clenshawcurtis.jl:12](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/clenshawcurtis.jl#L12)
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[FastTransforms/src/clenshawcurtis.jl:6](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/clenshawcurtis.jl#L6)
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[FastTransforms/src/clenshawcurtis.jl:6](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/clenshawcurtis.jl#L6)
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[FastTransforms/src/specialfunctions.jl:453](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L453)
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[FastTransforms/src/specialfunctions.jl:467](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L467)
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[FastTransforms/src/specialfunctions.jl:475](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L475)
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[FastTransforms/src/specialfunctions.jl:489](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L489)
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[FastTransforms/src/specialfunctions.jl:464](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L464)
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[FastTransforms/src/specialfunctions.jl:478](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L478)
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[FastTransforms/src/fejer.jl:7](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/fejer.jl#L7)
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[FastTransforms/src/fejer.jl:7](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/fejer.jl#L7)
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[FastTransforms/src/fejer.jl:12](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/fejer.jl#L12)
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[FastTransforms/src/fejer.jl:12](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/fejer.jl#L12)
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[FastTransforms/src/fejer.jl:21](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/fejer.jl#L21)
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[FastTransforms/src/fejer.jl:21](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/fejer.jl#L21)
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[FastTransforms/src/fejer.jl:26](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/fejer.jl#L26)
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[FastTransforms/src/fejer.jl:26](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/fejer.jl#L26)
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[FastTransforms/src/specialfunctions.jl:12](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L12)
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[FastTransforms/src/specialfunctions.jl:12](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L12)
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[FastTransforms/src/specialfunctions.jl:418](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L418)
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[FastTransforms/src/specialfunctions.jl:432](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L432)
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[FastTransforms/src/specialfunctions.jl:440](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L440)
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[FastTransforms/src/specialfunctions.jl:454](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L454)
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[FastTransforms/src/specialfunctions.jl:429](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L429)
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[FastTransforms/src/specialfunctions.jl:443](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L443)
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[FastTransforms/src/specialfunctions.jl:32](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L32)
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[FastTransforms/src/specialfunctions.jl:32](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L32)
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[FastTransforms/src/specialfunctions.jl:63](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L63)
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[FastTransforms/src/specialfunctions.jl:63](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L63)
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[FastTransforms/src/specialfunctions.jl:20](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L20)
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[FastTransforms/src/specialfunctions.jl:20](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L20)
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[FastTransforms/src/specialfunctions.jl:147](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L147)
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[FastTransforms/src/specialfunctions.jl:147](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L147)
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<a id="method__923.2" class="lexicon_definition"></a>
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#### Λ(x::Number) [](#method__923.2)
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#### Λ(z::Number) [](#method__923.2)
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The Lambda function Λ(z) = Γ(z+½)/Γ(z+1) for the ratio of gamma functions.
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[FastTransforms/src/specialfunctions.jl:141](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L141)
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[FastTransforms/src/specialfunctions.jl:141](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L141)
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---
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<a id="method__923.3" class="lexicon_definition"></a>
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#### Λ(z::Number, λ₁::Number, λ₂::Number) [](#method__923.3)
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The Lambda function Λ(z,λ₁,λ₂) = Γ(z+λ₁)/Γ(z+λ₂) for the ratio of gamma functions.
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*source:*
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[FastTransforms/src/specialfunctions.jl:160](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L160)
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[FastTransforms/src/specialfunctions.jl:26](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/e92c43c4bdbef278e459e31155eebe6c4f245429/src/specialfunctions.jl#L26)
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[FastTransforms/src/specialfunctions.jl:26](https://github.com/MikaelSlevinsky/FastTransforms.jl/tree/059ac0ba6b7b0216a19f5441660845326102abd1/src/specialfunctions.jl#L26)
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‎docs/api/index.md

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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(x::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)](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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‎docs/index.md

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1. N. Hale and A. Townsend. <a href="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. <a href="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. <a href="http://arxiv.org/abs/1604.07486">Fast polynomial transforms based on Toeplitz and Hankel matrices</a>, arXiv:1604.07486, 2016.

‎src/FastTransforms.jl

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jac2jac(x...)=th_jac2jac(x...)
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‎src/specialfunctions.jl

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"""
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The Lambda function Λ(z,λ₁,λ₂) = Γ(z+λ₁)/Γ(z+λ₂) for the ratio of gamma functions.
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"""
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Λ(z::Number,λ₁::Number,λ₂::Number) = exp(lgamma(z+λ₁)-lgamma(z+λ₂))
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function Λ(x::Float64,λ₁::Float64,λ₂::Float64)
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if min(x+λ₁,x+λ₂) ≥ 8.979120323411497
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exp(λ₂-λ₁+(x-.5)*log1p((λ₁-λ₂)/(x+λ₂)))*(x+λ₁)^λ₁/(x+λ₂)^λ₂*stirlingseries(x+λ₁)/stirlingseries(x+λ₂)
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else
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(x+λ₂)/(x+λ₁)*Λ(x+1.,λ₁,λ₂)
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end
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end
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Λ{T<:Number}(x::AbstractArray{T},λ₁::Number,λ₂::Number) = promote_type(T,typeof(λ₁),typeof(λ₂))[ Λ(x[i],λ₁,λ₂) for i in eachindex(x) ]
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Cnλ(n::Integer,λ::Float64) = 2^λ/sqrtpi*Λ(n+λ)
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Cnλ(n::Integer,λ::Number) = 2^λ/sqrt(oftype(λ,π))*Λ(n+λ)
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function Cnλ{T<:Integer}(n::UnitRange{T},λ::Number)

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