Special mathematical functions related to the beta and gamma functions.

```
beta(a, b)
lbeta(a, b)
```gamma(x)
lgamma(x)
psigamma(x, deriv = 0)
digamma(x)
trigamma(x)

choose(n, k)
lchoose(n, k)
factorial(x)
lfactorial(x)

a, b

non-negative numeric vectors.

x, n

numeric vectors.

k, deriv

integer vectors.

The functions `beta`

and `lbeta`

return the beta function
and the natural logarithm of the beta function,
$$B(a,b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}.$$
The formal definition is
$$B(a, b) = \int_0^1 t^{a-1} (1-t)^{b-1} dt$$
(Abramowitz and Stegun section 6.2.1, page 258). Note that it is only
defined in R for non-negative `a`

and `b`

, and is infinite
if either is zero.

The functions `gamma`

and `lgamma`

return the gamma function
\(\Gamma(x)\) and the natural logarithm of *the absolute value of* the
gamma function. The gamma function is defined by
(Abramowitz and Stegun section 6.1.1, page 255)
$$\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} dt$$
for all real `x`

except zero and negative integers (when
`NaN`

is returned). There will be a warning on possible loss of
precision for values which are too close (within about
\(10^{-8}\)) to a negative integer less than `-10`.

`factorial(x)`

(\(x!\) for non-negative integer `x`

)
is defined to be `gamma(x+1)`

and `lfactorial`

to be
`lgamma(x+1)`

.

The functions `digamma`

and `trigamma`

return the first and second
derivatives of the logarithm of the gamma function.
`psigamma(x, deriv)`

(`deriv >= 0`

) computes the
`deriv`

-th derivative of \(\psi(x)\).
$$\code{digamma(x)} = \psi(x) = \frac{d}{dx}\ln\Gamma(x) =
\frac{\Gamma'(x)}{\Gamma(x)}$$
\(\psi\) and its derivatives, the `psigamma()`

functions, are
often called the ‘polygamma’ functions, e.g.in
Abramowitz and Stegun (section 6.4.1, page 260); and higher
derivatives (`deriv = 2:4`

) have occasionally been called
‘tetragamma’, ‘pentagamma’, and ‘hexagamma’.

The functions `choose`

and `lchoose`

return binomial
coefficients and the logarithms of their absolute values. Note that
`choose(n, k)`

is defined for all real numbers \(n\) and integer
\(k\). For \(k \ge 1\) it is defined as
\(n(n-1)\cdots(n-k+1) / k!\),
as \(1\) for \(k = 0\) and as \(0\) for negative \(k\).
Non-integer values of `k`

are rounded to an integer, with a warning.
`choose(*, k)`

uses direct arithmetic (instead of
`[l]gamma`

calls) for small `k`

, for speed and accuracy
reasons. Note the function `combn`

(package
utils) for enumeration of all possible combinations.

The `gamma`

, `lgamma`

, `digamma`

and `trigamma`

functions are internal generic primitive functions: methods can be
defined for them individually or via the
`Math`

group generic.

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
*The New S Language*.
Wadsworth & Brooks/Cole. (For `gamma`

and `lgamma`

.)

Abramowitz, M. and Stegun, I. A. (1972)
*Handbook of Mathematical Functions*. New York: Dover.
https://en.wikipedia.org/wiki/Abramowitz_and_Stegun provides
links to the full text which is in public domain.
Chapter 6: Gamma and Related Functions.

`Arithmetic`

for simple, `sqrt`

for
miscellaneous mathematical functions and `Bessel`

for the
real Bessel functions.

For the incomplete gamma function see `pgamma`

.

# NOT RUN { require(graphics) choose(5, 2) for (n in 0:10) print(choose(n, k = 0:n)) factorial(100) lfactorial(10000) ## gamma has 1st order poles at 0, -1, -2, ... ## this will generate loss of precision warnings, so turn off op <- options("warn") options(warn = -1) x <- sort(c(seq(-3, 4, length.out = 201), outer(0:-3, (-1:1)*1e-6, "+"))) plot(x, gamma(x), ylim = c(-20,20), col = "red", type = "l", lwd = 2, main = expression(Gamma(x))) abline(h = 0, v = -3:0, lty = 3, col = "midnightblue") options(op) x <- seq(0.1, 4, length.out = 201); dx <- diff(x)[1] par(mfrow = c(2, 3)) for (ch in c("", "l","di","tri","tetra","penta")) { is.deriv <- nchar(ch) >= 2 nm <- paste0(ch, "gamma") if (is.deriv) { dy <- diff(y) / dx # finite difference der <- which(ch == c("di","tri","tetra","penta")) - 1 nm2 <- paste0("psigamma(*, deriv = ", der,")") nm <- if(der >= 2) nm2 else paste(nm, nm2, sep = " ==\n") y <- psigamma(x, deriv = der) } else { y <- get(nm)(x) } plot(x, y, type = "l", main = nm, col = "red") abline(h = 0, col = "lightgray") if (is.deriv) lines(x[-1], dy, col = "blue", lty = 2) } par(mfrow = c(1, 1)) ## "Extended" Pascal triangle: fN <- function(n) formatC(n, width=2) for (n in -4:10) { cat(fN(n),":", fN(choose(n, k = -2:max(3, n+2)))) cat("\n") } ## R code version of choose() [simplistic; warning for k < 0]: mychoose <- function(r, k) ifelse(k <= 0, (k == 0), sapply(k, function(k) prod(r:(r-k+1))) / factorial(k)) k <- -1:6 cbind(k = k, choose(1/2, k), mychoose(1/2, k)) ## Binomial theorem for n = 1/2 ; ## sqrt(1+x) = (1+x)^(1/2) = sum_{k=0}^Inf choose(1/2, k) * x^k : k <- 0:10 # 10 is sufficient for ~ 9 digit precision: sqrt(1.25) sum(choose(1/2, k)* .25^k) # }