Barnes's double $\Gamma$-function, $\gamma_{h}(0) = -\frac{1}{12}$?

I apologize if I may have gotten the name of the function incorrectly. The function $\gamma_h(x)$ is defined in Appendix A of the paper SEIBERG-WITTEN THEORY AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf by the relation $$\gamma_h(x+h) + \gamma_h(x-h) - 2\gamma_h(x) = \log\left(x\right).$$ For small $h$ we can write $\gamma_h(x) = \sum_{g=0}^\infty h^{2g - 2}\tilde\gamma_g(x)$. Using the relation above we found that $\tilde{\gamma}$ are determined uniquely up to linear terms in $x$ : $$\tilde\gamma_0(x) = \frac{1}{2}x^2\log\left(x\right) - \frac{3}{4}x^2, \ \tilde\gamma_1(x) = -\frac{1}{12}\log(x), \ \tilde\gamma_2 = -\frac{1}{240}\frac{1}{x^2},$$ $..., \tilde{\gamma}_g(x) = (B_{2g}/2g(2g - 2))(1/x^{2g-2}), ...$.

It seems from here that $\gamma_h(x)$ should approaches $\infty$ as $x\rightarrow 0$. However, the author claimed on the next page (page 79) that $\gamma_h(0) = -1/12$.

So I'm quite confused how can this be consistent? Does it mean that the formulas for $\tilde{\gamma}_g$ above (Equation (A.3) in the paper) is not valid for $x$ close to $0$? Any pointers to any references would be helpful, thank you.

• Just use (A.5). – Jon Apr 19 '18 at 13:13
• Is it right to say that the formula for $\tilde{\gamma}_g$ above only works for $x \rightarrow \infty, h \approx 0$ and I should take (A.5) as the definition? – user113988 Apr 20 '18 at 1:42
• I would so because this function is in strong need of regularization and (A.5) grants a finite result for $x\rightarrow 0$. – Jon Apr 20 '18 at 6:02