Gamma of integer plus one over integer

The gamma function satisfies

Γ(x+1) = x Γ(x)

and so in principle you could calculate the gamma function for any positive real number if you can calculate it on the interval (0, 1). For example,

$\begin{align*} \Gamma(\pi) &= (\pi - 1)\Gamma(\pi -1) \\ &= (\pi - 1)(\pi - 2) \Gamma(\pi - 2) \\ &= (\pi - 1)(\pi - 2) (\pi - 3)\Gamma(\pi - 3) \end{align*}$

So if you’re able to compute Γ(π-3) then you could compute Γ(π).

If n is a positive integer ε is some number between 0 and 1, is there a more direct way to express Γ(n + ε) in terms of Γ(ε)?

There is when ε is the reciprocal of an integer. For example,

$\begin{align*} \Gamma\left(n + \frac{1}{2}\right) &= \frac{(2n-1)!!}{2^n} \Gamma\left(\frac{1}{2}\right) \\ \Gamma\left(n + \frac{1}{3}\right) &= \frac{(3n-2)!!!}{3^n} \Gamma\left(\frac{1}{3}\right) \\ \Gamma\left(n + \frac{1}{4}\right) &= \frac{(4n-3)!!!!}{4^n} \Gamma\left(\frac{1}{4}\right) \\ \end{align*}$

The multiple exclamation marks may look strange if you haven’t seen this notation before. See the previous post for an explanation.

The general equation for integer k is

$\Gamma\left(n + \frac{1}{k}\right) = \frac{\Pi_k(kn - k + 1)}{k^n} \Gamma\left(\frac{1}{k}\right)$

where Π_k is a notation for k-factorial I suggested in the previous post. A more common notation would be to put !_(k) to the right of the argument rather than Π_k on the left.

Here’s Python code demonstrating that the equation above holds for randomly selected n and k.

from operator import mul
from functools import reduce
from scipy.special import gamma
import numpy as np

def multifactorial(n, k):
    return reduce(mul, range(n, 0, -k), 1)

def f1(n, k):
    return gamma(n + 1/k)

def f2(n, k):
    return multifactorial(k*n - k + 1, k)*gamma(1/k)/k**n

np.random.seed(20211014)
for _ in range(1000):
    n = np.random.randint(1, 50)
    k = np.random.randint(1, 50)    
    a, b = f1(n,k), f2(n, k)
    assert( abs(a/b - 1) <= 1e-10 )