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Bohr--Mollerup characterisation of the real Gamma function

Statement

Gamma is the unique positive log-convex function $f : (0, \infty) \to (0, \infty)$ with $f (1) = 1$ and $f (x + 1) = x f (x)$ .

Facts & Assumptions

Given: The real Gamma function and an arbitrary positive log-convex function $f$ satisfying the displayed normalization and recurrence.

[F1]

Every such function lies between common factorial bounds whose ratio is $(n + x) / n$ for $0 < x \leq 1$ (Log-convex solutions of the Gamma recurrence obey the Bohr--Mollerup factorial squeeze).

[F2]

For every $s > 0$ , $Γ (s + 1) = s Γ (s)$ , and $Γ (1) = 1$ (The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ ).

[F3]

The real Gamma function is strictly log-convex on $(0, \infty)$ (The real Gamma function is strictly log-convex).

[F4]

Every real lies in a unique half-open unit interval between consecutive integers (Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ ).

Proof

technique · direct

1.1

Gamma is positive by its Euler integrand, normalized and recurrent by [F2], and log-convex by [F3]. Thus it satisfies the characterizing properties.

F2F3

2.1

Fix $0 < x \leq 1$ . Apply [F1] to $f$ and to Gamma. Both lie between $G_{n} (x)$ and $((n + x) / n) G_{n} (x)$ for every $n \geq 2$ , and the ratio of these bounds tends to $1$ . The squeeze theorem therefore gives $f (x) = Γ (x)$ .

F1step 1.1algebra

3.1

By [F4], every positive real $y$ is an integer shift of a unique $x \in (0, 1]$ . Iterating the common recurrence from [F2] and the hypothesis on $f$ extends the equality of step 2.1 from that strip to $y$ .

step 2.1F2F4

4.1

Step 1.1 proves that Gamma has the properties, and steps 2.1 and 3.1 prove that every function with them equals Gamma. This is the claimed characterization.

step 1.1step 2.1step 3.1∎

Depends on

Used by

FALSE: normalization and the functional equation determine the real Gamma function False statement

Dependency tree · two levels

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Sources

K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 7 §4 (standard reference, not scraped)