Γ(1/n)

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If n is a positive integer, then rounding Γ(1/n) up to the nearest integer gives n. In symbols,

leftlceil Gammaleft( tfrac{1}{n}right) rightrceil = n

We an illustrate this with the following Python code.

>>> from scipy.special import gamma
>>> from math import ceil
>>> for n in range(1, 101):
    ... assert(ceil(gamma(1/n)) == n)

You can find a full proof in [1]. I’ll give a partial proof that may be more informative than the full proof.

The asymptotic expansion of the gamma function near zero is

Gamma(z) = frac{1}{z} - gamma + {cal O}(z^2)

where γ is the Euler-Mascheroni constant.

So when we set z = 1/n we find Γ(1/n) ≈ n − γ + O(1/n²). Since 0 < γ < 1, the theorem above is true for sufficiently large n. And it turns out “sufficiently large” can be replaced with n ≥ 1.

[1] Gamma at reciprocals of integers: 12225. American Mathematical Monthly. October 2022. pp 789–790.

The post Γ(1/n) first appeared on John D. Cook.

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