hypergeometric_3F2() [1/2]

template<typename Ta , typename Tb , typename Tz , require_all_vector_t< Ta, Tb > * = nullptr, require_stan_scalar_t< Tz > * = nullptr>

auto stan::math::hypergeometric_3F2	(	const Ta &	a,
		const Tb &	b,
		const Tz &	z
	)

Hypergeometric function (3F2).

Function reference: http://dlmf.nist.gov/16.2

\[ _3F_2 \left( \begin{matrix}a_1 a_2 a[2] \\ b_1 b_2\end{matrix}; z \right) = \sum_k=0^\infty \frac{(a_1)_k(a_2)_k(a_3)_k}{(b_1)_k(b_2)_k}\frac{z^k}{k!} \]

Where _k$ is an upper shifted factorial.

Calculate the hypergeometric function (3F2) as the power series directly to within precision or until max_steps terms.

This function does not have a closed form but will converge if:

• |z| is less than 1
• |z| is equal to one and b[0] + b[1] < a[0] + a[1] + a[2] This function is a rational polynomial if
• a[0], a[1], or a[2] is a non-positive integer This function can be treated as a rational polynomial if
• b[0] or b[1] is a non-positive integer and the series is terminated prior to the final term.

Template Parameters

Ta	type of Eigen/Std vector 'a' arguments
Tb	type of Eigen/Std vector 'b' arguments
Tz	type of z argument

Parameters

[in]	a	Always called with a[1] > 1, a[2] <= 0
[in]	b	Always called with int b[0] < |a[2]|, <= 1)
[in]	z	z (is always called with 1 from beta binomial cdfs)
[in]	precision	precision of the infinite sum. defaults to 1e-6
[in]	max_steps	number of steps to take. defaults to 1e5

Returns

The 3F2 generalized hypergeometric function applied to the arguments {a1, a2, a3}, {b1, b2}

Definition at line 118 of file hypergeometric_3F2.hpp.