Math.Hypergeometric

See Shaw, Ewart "Hypergeometric Functions and CDFs in J".

Documentation

hypergeometric

Arguments

:: (Ord a, Fractional a)

=> [a]

\( a_1,\ldots,a_p \)

-> [a]

\( b_1,\ldots,b_q \)

-> a

\( z \)

-> a

\( _pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \displaystyle\sum_{n=0}^\infty\frac{(a_1)_n\cdots(a_p)_n}{(b_1)_b\cdots(b_q)_n}\frac{z^n}{n!} \)

The radius of convergence is

\( \rho = \begin{cases} \infty & \text{if} & p<q+1 \\ 1 & \text{if} & p=q+1 \\ 0 & \text{if} & p>q+1 \\ \end{cases} \)

This iterates until the result stabilizes.

euler

Arguments

:: (Ord a, Floating a)

=> a

\(a\)

-> a

\(b\)

-> a

\(c\)

-> a

\(z\)

-> a

Euler's transform:

\( \displaystyle _2F_1(a,b;c;z) = (1-z)^{-a} {}_2F_1\left(a,c-b;c;\frac{z}{z-1}\right) \)

The right-hand side converges for all \(z\).

Koekoek, Roelef and Swarttouw, René F. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue.

Since: 0.1.5.0

erf :: (Ord a, Floating a) => a -> a

erf

ncdf :: (Ord a, Floating a) => a -> a

CDF of the standard normal \( N(0,1) \)