Abstract #
We derive new Hankel representations of select objects related to the Multiple Gamma Hierarchy, using the $K$-function as an auxiliary function. These Hankel forms enable further analysis through saddlepoint asymptotics.
\(K\)-Function Contour #
Recall that the \(K\)-function can be rewritten in terms of Hurwitz Zeta:
$$ \begin{aligned} &K(z) := \exp \left[ \zeta'(-1,z) - \zeta'(-1,1) \right]\ &\implies \log K(z) = \zeta'(-1,z) - \zeta'(-1,1) \end{aligned} $$By Boyadzhiev , the derivative of such Hurwitz Zeta function for \(\operatorname{Re}(z) >0\) can be expressed as:
$$ \zeta'(-1,z) = \frac{1}{2}\left(1-\gamma\right)\left(z^{2}-z+\frac{1}{6}\right)+\frac{1}{2\pi i}\int_{H}^{ }\frac{t^{-2}e^{zt}}{1-e^{t}}\operatorname{Log} (t) dt $$It is also known that \(\zeta'(-1,1)\) relates to the Glaisher-Kinkelin constant \(A\) per:
$$ \zeta'(-1,1) = \frac{1}{12} - \log A $$Applying both (2) and (3), we obtain:
$$ \begin{aligned} \log K(z) &= \left[ \frac{1}{2}\left(1-\gamma\right)\left(z^{2}-z+\frac{1}{6}\right)+\frac{1}{2\pi i}\int_{H}^{ }\frac{t^{-2}e^{zt}}{1-e^{t}}\operatorname{Log} (t) dt \right] - \left[ \frac{1}{12} - \log A \right] \ &= \left[ \frac{\left(1-\gamma \right)}{2}z^{2}+\frac{\left(\gamma-1\right)}{2}z+\left(\log A -\frac{\gamma}{12} \right) \right] + \frac{1}{2\pi i}\int_{H}^{ }\frac{t^{-2}e^{zt}}{1-e^{t}}\operatorname{Log} (t) dt \end{aligned} $$Log-Gamma Contour #
Lemma 1. *A quadratic \(az^2+bz+c\) can be expressed as a Hankel contour integral:
$$ \frac{1}{2\pi i} \int_H \frac{ct^2 + bt +2a}{t^3} e^{zt} dt $$*
Proof. We can extract the residues by Laurent coefficient \(a_{-1}\). Rewrite \(e^{zt}\) into its series form, and distribute. The \(\frac{1}{t}\) term emerges as \(az^2+bz+c\), which cancels the outer fraction by the Residue Theorem. ◻
The quadratic in (3) can thus be expressed as:
$$ \begin{aligned} \frac{1}{2\pi i}\int_{H}^{ }\frac{\left(\log A-\frac{\gamma}{12}\right)t^{2}+\left(\frac{\gamma-1}{2}\right)t+\left(1-\gamma\right)}{t^{3}}e^{zt}dt \end{aligned} $$Equation (1) can then be transformed into:
$$ \begin{aligned} \log K(z) &= \frac{1}{2\pi i}\int_{H}^{ }\frac{\left(\log A-\frac{\gamma}{12}\right)t^{2}+\left(\frac{\gamma-1}{2}\right)t+\left(1-\gamma\right)}{t^{3}}e^{zt}dt + \frac{1}{2\pi i}\int_{H}^{ }\frac{t^{-2}e^{zt}}{1-e^{t}}\operatorname{Log} (t) dt \ &= \frac{1}{2 \pi i} \int_H e^{zt} \left[ \frac{\left(\log A-\frac{\gamma}{12}\right)t^{2}+\left(\frac{\gamma-1}{2}\right)t+\left(1-\gamma\right)}{t^{3}}+ \frac{t^{-2} \operatorname{Log} (t)}{1-e^{t}} \right] dt \end{aligned} $$Representing Barnes G #
Lemma 2. The derivative of the Hurwitz Zeta function has Hankel contour representation:
*
Proof. Begin with the standard form of the Hurwitz Zeta:
$$ -\frac{\Gamma(1-s)}{2\pi i} \int_H \frac{(-t)^{s-1} e^{-at}}{1-e^{-t}} dt $$Differentiate with respect to \(s\): applying a product rule with differentiation under the integral. ◻
Corollary 3. The logarithm of the gamma function admits a simple method due to Lerch.
Proof. Take Lerch’s Lemma:
$$ \log\Gamma(z)=\zeta'(0,z)+\frac{1}{2}\log2\pi $$Substitute in the Hankel contour for the Hurwitz Zeta derivative:
$$ \begin{aligned} &\frac{1}{2\pi i} \left [\Gamma'(1) \left( \int_H \frac{(-t)^{-1} e^{-zt}}{1-e^{-t}}dt \right) - \Gamma(1)\left( \int_H \frac{(-t)^{-1}\operatorname{Log} (-t) e^{-zt}}{1-e^{-t}}dt \right) \right] + \frac{1}{2}\log2\pi \ &=\frac{1}{2\pi i} \left [ \int_H \frac{\gamma e^{-zt}}{t(1-e^{-t})}dt + \int_H \frac{\operatorname{Log} (-t)e^{-zt}}{t(1-e^{-t})}dt \right] + \frac{1}{2}\log2\pi \ &=\frac{1}{2\pi i} \left [ \int_H \frac{\gamma e^{-zt} + \operatorname{Log} (-t)e^{-zt}}{t(1-e^{-t})}dt \right] + \frac{1}{2}\log2\pi \ \end{aligned} $$Absorb the constant into the expression, and then factor \(e^{-zt}\):
$$ \begin{aligned} &\frac{1}{2\pi i} \left [ \int_H \frac{\gamma e^{-zt} + \operatorname{Log} (-t)e^{-zt}}{t(1-e^{-t})}dt \right] + \frac{1}{2\pi i} \int_H \frac{(\frac{1}{2} \log{2\pi})e^{-zt}}{t} dt\ &=\frac{1}{2\pi i} \left [ \int_H \frac{\gamma e^{-zt} + \operatorname{Log} (-t)e^{-zt}}{t(1-e^{-t})} + \frac{(\frac{1}{2} \log{2\pi})e^{-zt}}{t}dt \right] \ &=\frac{1}{2\pi i} \int_H e^{-zt}\left( \frac{\gamma + \operatorname{Log} (-t)}{t(1-e^{-t})} + \frac{(\frac{1}{2} \log{2\pi})}{t} \right)dt \end{aligned} $$Factoring \(e^{-zt}\) allows a nice expression for \(z\log\Gamma(z)\); if \(f(z)\) has contour integral \(\int_H e^{-zt}g(z)dz\), then \(zf(z)\) can be expressed as \(\int_H e^{-zt}g'(z)dz\), considering boundary decay. We then differentiate the inner expression to obtain the contour representation of \(z\log\Gamma(z)\).
$$ \begin{aligned} &z\log \Gamma(z) = \frac{1}{2\pi i} \int_H e^{-zt} D_t \left( \frac{\gamma + \operatorname{Log} (-t)}{t(1-e^{-t})} + \frac{(\frac{1}{2} \log{2\pi})}{t} \right)dt\ \end{aligned} $$◻
We proceed with the Barnes G and \(K\)-Function relationship, which holds for all \(z\). Divide both sides by \(K(z)\).
$$ \begin{aligned} & K(z) G(z) = e^{(z-1)\log{\Gamma(z)}}\ &\implies G(z) = \frac{e^{(z-1)\log{\Gamma(z)}}}{K(z)} \end{aligned} $$Proceed by taking the logarithm of both sides:
$$ \begin{aligned} \log G(z) &= \log \left( \frac{e^{(z-1)\log{\Gamma(z)}}}{K(z)} \right) \ & = z\log{\Gamma(z)} -\log\Gamma(z) - \log K(z) \end{aligned} $$We can now substitute our full expressions:
$$ \begin{aligned} \log G(z) = &\frac{1}{2\pi i} \int_H e^{-zt} D_t \left( \frac{\gamma + \operatorname{Log} (-t)}{t(1-e^{-t})} + \frac{(\frac{1}{2} \log{2\pi})}{t} \right)dt \ &- \frac{1}{2\pi i} \int_H e^{-zt}\left( \frac{\gamma + \operatorname{Log} (-t)}{t(1-e^{-t})} + \frac{(\frac{1}{2} \log{2\pi})}{t} \right)dt \ &- \frac{1}{2 \pi i} \int_H e^{zt} \left[ \frac{\left(\log A-\frac{\gamma}{12}\right)t^{2}+\left(\frac{\gamma-1}{2}\right)t+\left(1-\gamma\right)}{t^{3}}+ \frac{t^{-2} \operatorname{Log} (t)}{1-e^{t}} \right] dt \end{aligned} $$Log-Beta Contour #
The Multivariate Beta function can be defined as
$$ \mathrm{B}(\alpha_1,\alpha_2 \dots \alpha_n) = \frac{\Gamma(\alpha_1)\Gamma(\alpha_2) \dots \Gamma (\alpha_n)}{\Gamma{(\alpha_1+\alpha_2+\dots+\alpha_n)}} $$Corollary 4. *The logarithm of the Multivariate Beta function can be written as a Hankel contour integral:
$$ \log\mathrm{B}(\alpha_1,\alpha_2 \dots \alpha_n) =\frac{1}{2\pi i}\int_{H}^{ }\left(\sum_{k=1}^{n}e^{-a_{k}t}-e^{-\left(\sum_{k=1}^{n}a_{k}\right)t}\right)\left(\frac{\gamma+\operatorname{Log} (-t)}{t\left(1-e^{-t}\right)}+\frac{\frac{1}{2}\log 2\pi }{t}\right)dt $$*
Proof. Applying a logarithm scheme to the definition, we can observe that:
$$ \log\mathrm{B}(\alpha_1,\alpha_2 \dots \alpha_n) = \sum_{k=1}^{n}\log \Gamma(\alpha_k)\ - \log\Gamma{\left(\sum_{k=1}^n \alpha_k \right)} $$Substitute our derived form for \(\log\Gamma(z)\) into the first term. Then, repeat for the second term, treating \(\sum a_k\) as a parameter. After subtracting the contour integrals and distributing factors, the formula above emerges. ◻
References #
Khristo N. Boyadzhiev, “Evaluation of series with Hurwitz and Lerch zeta function coefficients by using Hankel contour integrals,” 2006. https://arxiv.org/abs/math/0606173
Eric W. Weisstein, “Glaisher-Kinkelin Constant,” From MathWorld—A Wolfram Resource. https://mathworld.wolfram.com/Glaisher-KinkelinConstant.html
Alexey Kuznetsov, “Computing the Barnes \(G\)-function and the gamma function in the entire complex plane,” 2022. https://arxiv.org/abs/2109.12061
Adrian Hernandez Vega, “Proof: Contour Integral for a General Quadratic,” 2026. https://adrianhv.com/work/quadraticcontour/