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## 3.13 Combinatorial Functions

real beta(real alpha, real beta)
Return the beta function applied to alpha and beta. The beta function, $$\text{B}(\alpha,\beta)$$, computes the normalizing constant for the beta distribution, and is defined for $$\alpha > 0$$ and $$\beta > 0$$. See section appendix for definition of $$\text{B}(\alpha, \beta)$$.

R beta(T1 x, T2 y)
Vectorized implementation of the beta function

real inc_beta(real alpha, real beta, real x)
Return the regularized incomplete beta function up to x applied to alpha and beta. See section appendix for a definition.

real lbeta(real alpha, real beta)
Return the natural logarithm of the beta function applied to alpha and beta. The beta function, $$\text{B}(\alpha,\beta)$$, computes the normalizing constant for the beta distribution, and is defined for $$\alpha > 0$$ and $$\beta > 0$$. $\text{lbeta}(\alpha,\beta) = \log \Gamma(a) + \log \Gamma(b) - \log \Gamma(a+b)$ See section appendix for definition of $$\text{B}(\alpha, \beta)$$.

R lbeta(T1 x, T2 y)
Vectorized implementation of the lbeta function

R tgamma(T x)
gamma function applied to x. The gamma function is the generalization of the factorial function to continuous variables, defined so that $$\Gamma(n+1) = n!$$. See for a full definition of $$\Gamma(x)$$. The function is defined for positive numbers and non-integral negative numbers,

R lgamma(T x)
natural logarithm of the gamma function applied to x,

R digamma(T x)
digamma function applied to x. The digamma function is the derivative of the natural logarithm of the Gamma function. The function is defined for positive numbers and non-integral negative numbers

R trigamma(T x)
trigamma function applied to x. The trigamma function is the second derivative of the natural logarithm of the Gamma function

real lmgamma(int n, real x)
Return the natural logarithm of the multivariate gamma function $$\Gamma_n$$ with n dimensions applied to x. $\text{lmgamma}(n,x) = \begin{cases} \frac{n(n-1)}{4} \log \pi + \sum_{j=1}^n \log \Gamma\left(x + \frac{1 - j}{2}\right) & \text{if } x\not\in \{\dots,-3,-2,-1,0\}\\ \textrm{error} & \text{otherwise} \end{cases}$

R lmgamma(T1 x, T2 y)
Vectorized implementation of the lmgamma function

real gamma_p(real a, real z)
Return the normalized lower incomplete gamma function of a and z defined for positive a and nonnegative z. $\mathrm{gamma\_p}(a,z) = \begin{cases} \frac{1}{\Gamma(a)}\int_0^zt^{a-1}e^{-t}dt & \text{if } a > 0, z \geq 0 \\ \textrm{error} & \text{otherwise} \end{cases}$

R gamma_p(T1 x, T2 y)
Vectorized implementation of the gamma_p function

real gamma_q(real a, real z)
Return the normalized upper incomplete gamma function of a and z defined for positive a and nonnegative z. $\mathrm{gamma\_q}(a,z) = \begin{cases} \frac{1}{\Gamma(a)}\int_z^\infty t^{a-1}e^{-t}dt & \text{if } a > 0, z \geq 0 \\[6pt] \textrm{error} & \text{otherwise} \end{cases}$

R gamma_q(T1 x, T2 y)
Vectorized implementation of the gamma_q function

real binomial_coefficient_log(real x, real y)
Warning: This function is deprecated and should be replaced with lchoose. Return the natural logarithm of the binomial coefficient of x and y. For non-negative integer inputs, the binomial coefficient function is written as $$\binom{x}{y}$$ and pronounced “x choose y.” This function generalizes to real numbers using the gamma function. For $$0 \leq y \leq x$$, $\mathrm{binomial\_coefficient\_log}(x,y) = \log\Gamma(x+1) - \log\Gamma(y+1) - \log\Gamma(x-y+1).$

R binomial_coefficient_log(T1 x, T2 y)
Vectorized implementation of the binomial_coefficient_log function

int choose(int x, int y)
Return the binomial coefficient of x and y. For non-negative integer inputs, the binomial coefficient function is written as $$\binom{x}{y}$$ and pronounced “x choose y.” In its the antilog of the lchoose function but returns an integer rather than a real number with no non-zero decimal places. For $$0 \leq y \leq x$$, the binomial coefficient function can be defined via the factorial function $\text{choose}(x,y) = \frac{x!}{\left(y!\right)\left(x - y\right)!}.$

R choose(T1 x, T2 y)
Vectorized implementation of the choose function

real bessel_first_kind(int v, real x)
Return the Bessel function of the first kind with order v applied to x. $\mathrm{bessel\_first\_kind}(v,x) = J_v(x),$ where $J_v(x)=\left(\frac{1}{2}x\right)^v \sum_{k=0}^\infty \frac{\left(-\frac{1}{4}x^2\right)^k}{k!\, \Gamma(v+k+1)}$

R bessel_first_kind(T1 x, T2 y)
Vectorized implementation of the bessel_first_kind function

real bessel_second_kind(int v, real x)
Return the Bessel function of the second kind with order v applied to x defined for positive x and v. For $$x,v > 0$$, $\mathrm{bessel\_second\_kind}(v,x) = \begin{cases} Y_v(x) & \text{if } x > 0 \\ \textrm{error} & \text{otherwise} \end{cases}$ where $Y_v(x)=\frac{J_v(x)\cos(v\pi)-J_{-v}(x)}{\sin(v\pi)}$

R bessel_second_kind(T1 x, T2 y)
Vectorized implementation of the bessel_second_kind function

real modified_bessel_first_kind(int v, real z)
Return the modified Bessel function of the first kind with order v applied to z defined for all z and v. $\mathrm{modified\_bessel\_first\_kind}(v,z) = I_v(z)$ where ${I_v}(z) = \left(\frac{1}{2}z\right)^v\sum_{k=0}^\infty \frac{\left(\frac{1}{4}z^2\right)^k}{k!\Gamma(v+k+1)}$

R modified_bessel_first_kind(T1 x, T2 y)
Vectorized implementation of the modified_bessel_first_kind function

real modified_bessel_second_kind(int v, real z)
Return the modified Bessel function of the second kind with order v applied to z defined for positive z and v. $\mathrm{modified\_bessel\_second\_kind}(v,z) = \begin{cases} K_v(z) & \text{if } z > 0 \\ \textrm{error} & \text{if } z \leq 0 \end{cases}$ where ${K_v}(z) = \frac{\pi}{2}\cdot\frac{I_{-v}(z) - I_{v}(z)}{\sin(v\pi)}$

R modified_bessel_second_kind(T1 x, T2 y)
Vectorized implementation of the modified_bessel_second_kind function

real falling_factorial(real x, real n)
Return the falling factorial of x with power n defined for positive x and real n. $\mathrm{falling\_factorial}(x,n) = \begin{cases} (x)_n & \text{if } x > 0 \\ \textrm{error} & \text{if } x \leq 0 \end{cases}$ where $(x)_n=\frac{\Gamma(x+1)}{\Gamma(x-n+1)}$

R falling_factorial(T1 x, T2 y)
Vectorized implementation of the falling_factorial function

real lchoose(real x, real y)
Return the natural logarithm of the generalized binomial coefficient of x and y. For non-negative integer inputs, the binomial coefficient function is written as $$\binom{x}{y}$$ and pronounced “x choose y.” This function generalizes to real numbers using the gamma function. For $$0 \leq y \leq x$$, $\mathrm{binomial\_coefficient\_log}(x,y) = \log\Gamma(x+1) - \log\Gamma(y+1) - \log\Gamma(x-y+1).$

real log_falling_factorial(real x, real n)
Return the log of the falling factorial of x with power n defined for positive x and real n. $\mathrm{log\_falling\_factorial}(x,n) = \begin{cases} \log (x)_n & \text{if } x > 0 \\ \textrm{error} & \text{if } x \leq 0 \end{cases}$

real rising_factorial(real x, int n)
Return the rising factorial of x with power n defined for positive x and integer n. $\mathrm{rising\_factorial}(x,n) = \begin{cases} x^{(n)} & \text{if } x > 0 \\ \textrm{error} & \text{if } x \leq 0 \end{cases}$ where $x^{(n)}=\frac{\Gamma(x+n)}{\Gamma(x)}$

R rising_factorial(T1 x, T2 y)
Vectorized implementation of the rising_factorial function

real log_rising_factorial(real x, real n)
Return the log of the rising factorial of x with power n defined for positive x and real n. $\mathrm{log\_rising\_factorial}(x,n) = \begin{cases} \log x^{(n)} & \text{if } x > 0 \\ \textrm{error} & \text{if } x \leq 0 \end{cases}$

R log_rising_factorial(T1 x, T2 y)
Vectorized implementation of the log_rising_factorial function