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Generalized binomial coefficient #293

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66e7560
function definition
Jan 13, 2021
bb06207
WIP documentation
Jan 13, 2021
c89743a
extending Base.binomial
Jan 15, 2021
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roadmap for tests
Jan 15, 2021
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tests
Jan 16, 2021
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documentation
Jan 16, 2021
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1 change: 1 addition & 0 deletions docs/src/functions_list.md
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Expand Up @@ -51,6 +51,7 @@ SpecialFunctions.besseli
SpecialFunctions.besselix
SpecialFunctions.besselk
SpecialFunctions.besselkx
SpecialFunctions.binomial
SpecialFunctions.jinc
SpecialFunctions.ellipk
SpecialFunctions.ellipe
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1 change: 1 addition & 0 deletions docs/src/functions_overview.md
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Expand Up @@ -22,6 +22,7 @@ Here the *Special Functions* are listed according to the structure of [NIST Digi
| [`logbeta(x,y)`](@ref SpecialFunctions.logbeta) | accurate `log(beta(x,y))` for large `x` or `y` |
| [`logabsbeta(x,y)`](@ref SpecialFunctions.logabsbeta) | accurate `log(abs(beta(x,y)))` for large `x` or `y` |
| [`logabsbinomial(x,y)`](@ref SpecialFunctions.logabsbinomial) | accurate `log(abs(binomial(n,k)))` for large `n` and `k` near `n/2` |
| [`binomial(x,y)`](@ref SpecialFunctions.binomial) | generalized binomial coefficient ``{ x \choose y}`` for ``x,y \in \mathbb{C}``|


## [Exponential and Trigonometric Integrals](https://dlmf.nist.gov/6)
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1 change: 1 addition & 0 deletions src/SpecialFunctions.jl
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Expand Up @@ -29,6 +29,7 @@ export
bessely0,
bessely1,
besselyx,
binomial,
jinc,
dawson,
ellipk,
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19 changes: 19 additions & 0 deletions src/gamma.jl
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Expand Up @@ -926,3 +926,22 @@ function logabsbinomial(n::T, k::T) where {T<:Integer}
end
end
logabsbinomial(n::Integer, k::Integer) = logabsbinomial(promote(n, k)...)


# this trickery is needed while the deprecated method in Base exists
@static if !hasmethod(Base.binomial, Tuple{Number, Number})
import Base: binomial
end
binomial(n, k) = Base.binomial(n, k) # to make SpecialFunctions.binomial work unconditionally

"""
binomial(x, y)

Generalized binomial coefficient
``{x \\choose y} = \\frac{1}{(x+1) \\Beta(x-y+1,y+1)}``.

External links: [Wikipedia](https://en.wikipedia.org/wiki/Binomial_coefficient#Two_real_or_complex_valued_arguments)

See also [`beta(a,b)`](@ref SpecialFunctions.beta).
"""
binomial(x::Number, y::Number) = inv((x+1) * beta(x-y+1, y+1))