grad_pFq()

template<bool calc_a = true, bool calc_b = true, bool calc_z = true, typename TpFq , typename Ta , typename Tb , typename Tz , typename T_Rtn = return_type_t<Ta, Tb, Tz>, typename Ta_Rtn = promote_scalar_t<T_Rtn, plain_type_t<Ta>>, typename Tb_Rtn = promote_scalar_t<T_Rtn, plain_type_t<Tb>>>

std::tuple< Ta_Rtn, Tb_Rtn, T_Rtn > stan::math::grad_pFq ( const TpFq &  pfq_val, const Ta &  a, const Tb &  b, const Tz &  z, double  precision = 1e-14, int  max_steps = 1e6  )

inline

Returns the gradient of generalized hypergeometric function wrt to the input arguments: \( _pF_q(a_1,...,a_p;b_1,...,b_q;z) \).

Where: \( \frac{\partial }{\partial a_1} = \sum_{k=0}^{\infty}{ \frac {\psi\left(k+a_1\right)\left(\prod_{j=1}^p\left(a_j\right)_k\right)z^k} {k!\prod_{j=1}^q\left(b_j\right)_k}} - \psi\left(a_1\right){}_pF_q(a_1,...,a_p;b_1,...,b_q;z) \) \( \frac{\partial }{\partial b_1} = \psi\left(b_1\right){}_pF_q(a_1,...,a_p;b_1,...,b_q;z) - \sum_{k=0}^{\infty}{ \frac {\psi\left(k+b_1\right)\left(\prod_{j=1}^p\left(a_j\right)_k\right)z^k} {k!\prod_{j=1}^q\left(b_j\right)_k}} \)

\( \frac{\partial }{\partial z} = \frac{\prod_{j=1}^{p}(a_j)}{\prod_{j=1}^{q} (b_j)}\ {}_pF_q(a_1+1,...,a_p+1;b_1+1,...,b_q+1;z) \)

Noting the the recurrence relation for the digamma function: \( \psi(x + 1) = \psi(x) + \frac{1}{x} \), the gradients for the function with respect to a and b then simplify to: \( \frac{\partial }{\partial a_1} = \sum_{k=1}^{\infty}{ \frac {\left(1 + \sum_{m=0}^{k-1}\frac{1}{m+a_1}\right) * \left(\prod_{j=1}^p\left(a_j\right)_k\right)z^k} {k!\prod_{j=1}^q\left(b_j\right)_k}} - {}_pF_q(a_1,...,a_p;b_1,...,b_q;z) \) \( \frac{\partial }{\partial b_1} = {}_pF_q(a_1,...,a_p;b_1,...,b_q;z) - \sum_{k=1}^{\infty}{ \frac {\left(1 + \sum_{m=0}^{k-1}\frac{1}{m+b_1}\right) * \left(\prod_{j=1}^p\left(a_j\right)_k\right)z^k} {k!\prod_{j=1}^q\left(b_j\right)_k}} \)

Template Parameters

calc_a	Boolean for whether to calculate derivatives wrt to 'a'
calc_b	Boolean for whether to calculate derivatives wrt to 'b'
calc_z	Boolean for whether to calculate derivatives wrt to 'z'
TpFq	Scalar type of hypergeometric_pFq return value
Ta	Eigen type with either one row or column at compile time
Tb	Eigen type with either one row or column at compile time
Tz	Scalar type

Parameters

[in]	pfq_val	Return value from the hypergeometric_pFq function
[in]	a	Vector of 'a' arguments to function
[in]	b	Vector of 'b' arguments to function
[in]	z	Scalar z argument
[in]	precision	Convergence criteria for infinite sum
[in]	max_steps	Maximum number of iterations

Returns

Tuple of gradients

Definition at line 92 of file grad_pFq.hpp.