1 #ifndef STAN_MATH_PRIM_SCAL_FUN_RISING_FACTORIAL_HPP
2 #define STAN_MATH_PRIM_SCAL_FUN_RISING_FACTORIAL_HPP
3
4 #include <boost/math/special_functions/factorials.hpp>
5 #include <stan/math/prim/scal/fun/boost_policy.hpp>
6 #include <stan/math/prim/scal/err/check_not_nan.hpp>
7 #include <stan/math/prim/scal/err/check_nonnegative.hpp>
8 #include <limits>
9
10 namespace stan {
11 namespace math {
12
13 /**
14 * Return the rising factorial function evaluated
15 * at the inputs.
16 * Will throw for NaN x and for negative n
17 *
18 * @tparam T Type of x argument.
19 * @param x Argument.
20 * @param n Argument
21 * @return Result of rising factorial function.
22 * @throw std::domain_error if x is NaN
23 * @throw std::domain_error if n is negative
24 *
25 \f[
26 \mbox{rising\_factorial}(x, n) =
27 \begin{cases}
28 \textrm{error} & \mbox{if } x \leq 0\\
29 x^{(n)} & \mbox{if } x > 0 \textrm{ and } -\infty \leq n \leq \infty \\[6pt]
30 \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n = \textrm{NaN}
31 \end{cases}
32 \f]
33
34 \f[
35 \frac{\partial\, \mbox{rising\_factorial}(x, n)}{\partial x} =
36 \begin{cases}
37 \textrm{error} & \mbox{if } x \leq 0\\
38 \frac{\partial\, x^{(n)}}{\partial x} & \mbox{if } x > 0 \textrm{ and } -\infty
39 \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n =
40 \textrm{NaN} \end{cases} \f]
41
42 \f[
43 \frac{\partial\, \mbox{rising\_factorial}(x, n)}{\partial n} =
44 \begin{cases}
45 \textrm{error} & \mbox{if } x \leq 0\\
46 \frac{\partial\, x^{(n)}}{\partial n} & \mbox{if } x > 0 \textrm{ and } -\infty
47 \leq n \leq \infty \\[6pt] \textrm{NaN} & \mbox{if } x = \textrm{NaN or } n =
48 \textrm{NaN} \end{cases} \f]
49
50 \f[
51 x^{(n)}=\frac{\Gamma(x+n)}{\Gamma(x)}
52 \f]
53
54 \f[
55 \frac{\partial \, x^{(n)}}{\partial x} = x^{(n)}(\Psi(x+n)-\Psi(x))
56 \f]
57
58 \f[
59 \frac{\partial \, x^{(n)}}{\partial n} = (x)_n\Psi(x+n)
60 \f]
61 *
62 */
63 template <typename T>
rising_factorial(const T & x,int n)64 inline typename boost::math::tools::promote_args<T>::type rising_factorial(
65 const T& x, int n) {
66 static const char* function = "rising_factorial";
67 check_not_nan(function, "first argument", x);
68 check_nonnegative(function, "second argument", n);
69 return boost::math::rising_factorial(x, n, boost_policy_t());
70 }
71 } // namespace math
72 } // namespace stan
73 #endif
74