lmgamma() [2/4]

template<typename T , require_arithmetic_t< T > * = nullptr>

return_type_t< T > stan::math::lmgamma ( int  k, T  x  )

inline

Return the natural logarithm of the multivariate gamma function with the specified dimensions and argument.

The multivariate gamma function $\Gamma_k(x)$ for dimensionality $k$ and argument $x$ is defined by

$\Gamma_k(x) = \pi^{k(k-1)/4} \, \prod_{j=1}^k \Gamma(x + (1 - j)/2)$,

where $\Gamma()$ is the gamma function.

$$\mbox{lmgamma}(n, x) = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \ln\Gamma_n(x) & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$$

$$\frac{\partial\, \mbox{lmgamma}(n, x)}{\partial x} = \begin{cases} \textrm{error} & \mbox{if } x\in \{\dots, -3, -2, -1, 0\}\\ \frac{\partial\, \ln\Gamma_n(x)}{\partial x} & \mbox{if } x\not\in \{\dots, -3, -2, -1, 0\}\\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN} \end{cases}$$

$$\ln\Gamma_n(x) = \pi^{n(n-1)/4} \, \prod_{j=1}^n \Gamma(x + (1 - j)/2)$$

$$\frac{\partial \, \ln\Gamma_n(x)}{\partial x} = \sum_{j=1}^n \Psi(x + (1 - j) / 2)$$

Template Parameters

T	type of scalar

Parameters

k	Number of dimensions.
x	Function argument.

Returns

Natural log of the multivariate gamma function.

Definition at line 55 of file lmgamma.hpp.