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Special mathematical functions in Julia

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Chain rules for partial derivatives of incomplete beta. Code is by @arzwa and taken from https://github.com/arzwa/IncBetaDer/blob/main/src/beta_inc_grad.jl Related to effort to provide julia implementations of pdf/cdf like functions (https://github.com/JuliaStats/StatsFuns.jl/pull/113) and associated...

It would be good (to go with #364 ) to have a 1.x release that include the deprecation for `factorial(::Number)`, which was removed in https://github.com/JuliaMath/SpecialFunctions.jl/pull/297

![image](https://user-images.githubusercontent.com/10637482/142762911-ff95b1a7-e3fb-4ef8-b508-19b0623f59ac.png)

This PR uses `RealDot.realdot` to perform computations of `real(dot(x, y))` more efficiently. [RealDot.jl](https://github.com/JuliaMath/RealDot.jl) is a very lightweight package that only defines `realdot` and only depends on LinearAlgebra: https://github.com/JuliaMath/RealDot.jl/blob/main/src/RealDot.jl

This PR adds special handling for arguments of 1 in `beta` and `logabsbeta`. This fixes one of the issues in https://github.com/JuliaStats/StatsFuns.jl/pull/126 (I checked locally that the problem is fixed) but...

It was noted that `gamma` and `lgamma` did not handle `Float16` in #131, which was subsequently fixed in #132. I'm opening this issue to track correct handling in the remaining...

What the title says. The problem seems to be with the Newton's method implementation in lines 432-437 of `erf.jl`. ```julia while true # Newton iterations Δx = sqrtπhalf * (erf(x)...

Copied from https://github.com/JuliaMath/SpecialFunctions.jl/pull/305: > I added ChainRules definitions for gamma(a, x), loggamma(a, x), and gamma_inc according to https://functions.wolfram.com/GammaBetaErf/GammaRegularized/introductions/Gammas/ShowAll.html . Similar to the Bessel functions, derivatives with respect to the first...

Hi! I'm using the besselj and bessely ( )functions on a function to solve their zeros, as is shown in the source code of boostlibrary (`https://github.com/Ceboc/BesselZeros`). As i want to...

enhancement

As stated [here](https://github.com/JuliaMath/SpecialFunctions.jl/blob/master/src/gamma.jl#L888), if `x` is a float, instead of `factorial(x)` it should use the `gamma(x+1)` function instead. This is not the what is happening, as seen by this output:...