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Log Gamma ratio series on the unit disk

Statement

For $∣ w ∣ < 1$ , choose the analytic logarithm equal to $0$ at $w = 0$ . Then $\log \frac{Γ (1 - w)}{Γ (1 + w)} = 2 γ w + 2 \sum_{m = 1}^{\infty} \frac{ζ (2 m + 1)}{2 m + 1} w^{2 m + 1} .$ The series converges locally uniformly on the unit disk.

Facts & Assumptions

Given: $w \in C$ , $∣ w ∣ < 1$ ; $γ$ and $ζ (s)$ have the definitions in the cited mathematical suppliers.

[F1]

The Weierstrass product for reciprocal Gamma: $1 / Γ (1 + w) = e^{γ w} \prod_{n \geq 1} (1 + w / n) e^{- w / n}$ , locally uniformly, and the corresponding formula with $w$ replaced by $- w$ .

[F2]

The Euler-Mascheroni constant defines $γ$ ; The Riemann zeta function on the half-plane $Re s > 1$ gives $ζ (s) = \sum_{n \geq 1} n^{- s}$ for $Re s > 1$ .

Proof

1.1givenF1algebra

Fix $0 < r < 1$ and $∣ w ∣ \leq r$ . For each $n \geq 1$ , the power-series logarithms at $1$ are defined because $∣ w / n ∣ < 1$ , and $\log (1 + w / n) - \log (1 - w / n) - 2 w / n = 2 \sum_{m \geq 1} \frac{w^{2 m + 1}}{(2 m + 1) n^{2 m + 1}} .$ The absolute value of the right side is at most $2 r^{3} (1 - r^{2})^{- 1} n^{- 3}$ , since $2 m + 1 \geq 3$ and the geometric tail is bounded by that expression. Thus the sum over $n$ converges uniformly on $∣ w ∣ \leq r$ .

2.1F1F2step 1.1algebra∎

Taking the quotient of the two products in [F1] and using the logarithm branch fixed at zero gives $\log \frac{Γ (1 - w)}{Γ (1 + w)} = 2 γ w + \sum_{n \geq 1} (\log (1 + w / n) - \log (1 - w / n) - 2 w / n) .$ The bound in 1.1 justifies interchanging the two sums locally uniformly. Summing $n^{- (2 m + 1)}$ by [F2] gives the stated series. Since every compact subset of $∣ w ∣ < 1$ lies in some $∣ w ∣ \leq r < 1$ , the convergence is locally uniform throughout the disk.

Remarks

The expansion is an exact local analytic identity. In an on-shell four-graviton amplitude the variables $s, t, u$ are dimensionless; the kinematic relation $s + t + u = 0$ cancels the linear term after summing the three copies.

Depends on

Dependency tree · 2 levels

Sources

Eric D’Hoker and Michael B. Green, Exploring transcendentality in superstring amplitudes