Log Rising Factorial (Pochhammer Symbol)

Computes the logarithm of the rising factorial (Pochhammer symbol) \((\alpha)_J = \alpha(\alpha+1)\cdots(\alpha+J-1)\).

Usage

log_rising_factorial(alpha, J)

Arguments

alpha

Numeric; concentration parameter (must be positive scalar).

J

Integer; number of terms (must be >= 1).

Value

Numeric; \(\log(\alpha)_J\).

Details

Uses the identity: $$(\alpha)_J = \frac{\Gamma(\alpha+J)}{\Gamma(\alpha)}$$

Therefore: $$\log(\alpha)_J = \log\Gamma(\alpha+J) - \log\Gamma(\alpha)$$

This is numerically stable for all \(\alpha > 0\) and \(J \geq 1\).

See also

lgamma for the log-gamma function

Examples

# (2)_3 = 2 * 3 * 4 = 24
exp(log_rising_factorial(2, 3))
#> [1] 24

# (1)_5 = 5!
exp(log_rising_factorial(1, 5))
#> [1] 120
factorial(5)
#> [1] 120