Abstract #
Vigneras (1979) proved the existence and recursive properties of the Multi Gamma Hierarchy, as well as a general version of the Bohr-Mollerup theorem for discrete cases. In this project, we extend the hierarchy to fractional members, proving their existence and asymptotic behavior with the Riemann-Liouville integral. Furthermore, we derive closed expressions through differentiation of the Hurwitz zeta function, showing they satisfy the Bohr-Mollerup theorem for continuous cases as well. This completes the hierarchy for non-negative fractional indices, while opening the way for further study in complex-valued indices.
Definitions #
Definition 1 (Gamma Function). The gamma function is the analytic continuation of the factorial, uniquely characterized by the Bohr-Mollerup Criteria: recurrence through \(\Gamma (z)=z\Gamma (z-1)\), log-convexity, and \(\Gamma (1)=1\).
Definition 2 (Axioms of the Vignéras Hierarchy). Each function in the Vignéras hierarchy is uniquely and recursively characterized by the following three requirements:
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Functional Equation: \(G_n (z+1) = G_{n-1}(z) G_n(z) \forall z>0\)
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Normalization: \(G_n(1) = 1, \forall n \geq 1\)
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Positive Derivative: \(D^{n+1} \log G_n(z+1) \geq 0, \forall z\geq 0\).
Definition 3 (Big Theta \(\Theta\)). *A function is said to be in \(\Theta(g(n))\) if and only if there exist positive constants such that:
$$ 0 \leq c_1 g(n) \leq f(n) \leq c_2 g(n) $$Definition 4 (Hankel Contour Integral). A Hankel Contour Integral \(\oint _H f(z) dz\) evaluates \(f(z)\) around a contour \(H\) from \(-\varepsilon, \infty\) to \(\infty, \varepsilon\).
Member Asymptotics #
Theorem 5 (Leading Asymptotic of the Vignéras Multi-Gamma Functions). The Leading Asymptotic of the \(\log G_n(x)\) has complexity \(\Theta(x^n \log x)\).
Proof. We proceed by way of mathematical induction, showing the base case (\(n=1\)) and then proving for general \(n\) with \(n-1\) as inductive hypothesis.
Base Step: \(n=1\)
Gamma is defined to be the first case of the Vignéras hierarchy; using Stirling’s approximation and \(\Gamma(x) = (x-1)!\),
$$ \Gamma(x) \sim \sqrt{2\pi} (x-1)^{x-\frac{1}{2}} e^{-(x-1)} $$Taking logarithms,
$$ \log \Gamma(x) \sim \log(\sqrt{2\pi}) + \left(x-\tfrac{1}{2}\right)\log(x-1) - (x-1) $$Since \(\log(x-1) = \log x + o(1)\), it follows that
$$ \begin{aligned} \log \Gamma(x) \sim x\log x - x. \\ \implies \log G_1(x) \in \Theta(x^1\log x) \end{aligned} $$Inductive Step: \(n-1\)
Assume as inductive hypothesis that \(\log G_{n-1}(x)\) has complexity \(\Theta(x^{n-1} \log x)\).
A level \(G_n\) can be written as a finite product of the previous level.
Taking the logarithm allows us to convert the right into summations:
$$ \begin{aligned} & \log\left(G_{n}\left(x+1\right)\right)=\sum_{k=1}^{x-1}\log\left(G_{n-1}(k)\right) \end{aligned} $$Substitute the summand with the inductive hypothesis and rewrite in terms of an integral:
$$ \begin{aligned} \log\left(G_{n}\left(x+1\right)\right)& =\sum_{k=1}^{x-1}\log\left(G_{n-1}\left(k\right)\right) \\ & \sim\sum_{k=1}^{x-1}k^{n-1}\log k \\ & = \int_{1}^{x-1}\left(k^{n-1}\log k\right)dk+\text{o}\left(k^{n-1}\log k\right)\\ \end{aligned} $$With Integration by Parts, the resulting integral can be written as:
$$ \begin{aligned} & \left. \left[ \frac{k^{n}\left(n\log\left(k\right)-1\right)}{n^{2}} \right] \right|_{1}^{x-1} + \text{o}\left(k^{n-1}\log k\right) \\ &=\frac{\left(x-1\right)^{n}\left(n\log\left(x-1\right)-1\right)}{n^{2}}-\frac{\left(n\log1-1\right)}{n^{2}} + \text{o}\left(k^{n-1}\log k\right) \\ &=\frac{\left(x-1\right)^{n}\left(n\log\left(x-1\right)-1\right)}{n^{2}}+\frac{1}{n^{2}} + \text{o}\left(k^{n-1}\log k\right) \\ \end{aligned} $$Since the only remaining term is \(\text{o}\left(k^{n-1}\log k\right)\), which must grow slower than \(k^n\log k\), the complexity of \(G_n\) must be \(\Theta (x^n \log x)\) ◻
Fractional Member Asymptotics #
Theorem 6 (Leading Asymptotic of the Fractional Vignéras Multi-Gamma Functions). The Log of the Fractional Vignéras \(\log G_\alpha(x)\) has complexity \(\Theta(x^\alpha \log x)\).
Proof. Under the Riemann–Liouville fractional integral, scaling \(t = xu\) gives
$$ J^\alpha f(x) = x^\alpha \int_0^1 (1-u)^{\alpha-1} f(xu)\,du. $$For \(f(x)=x^a\log x\), substitution yields
$$ f(xu)=x^a u^a(\log x + \log u). $$Thus
$$ J^\alpha(x^a\log x) = x^{a+\alpha}\left(C_1 \log x + C_2\right), $$where \(C_1,C_2\) depend only on \(\alpha,a\). The dominant growth scales with the exponent.
$$ \Theta(x^{a+\alpha}\log x). $$◻
Integral Form #
Theorem 7 (Formula for the Fractional Vignéras Hierarchy Member). For \(\Re(z) > 0\):
$$ \begin{aligned} \log G_a(z) &= \frac{\partial}{\partial s} \left. \frac{\Gamma(1-s)}{2\pi i} \oint_H \frac{e^{-zt}(-t)^{s-1}}{(1-e^{-t})^{a}} \, dt \right|_{s=0} - C_a \\ \implies G_a(z) &= \exp\!\left( \frac{\partial}{\partial s} \left. \frac{\Gamma(1-s)}{2\pi i} \oint_H \frac{e^{-zt}(-t)^{s-1}}{(1-e^{-t})^{a}} \, dt \right|_{s=0} - C_a \right) \end{aligned} $$Reccurence Functional Equation #
Proof. Apply a logarithm to both sides of the recurrence:
$$ \begin{aligned} &G_n(z+1) = G_{n-1}(z)G_n(z) \\ &\implies \log G_n(z+1) - \log G_n(z) = \log G_{n-1}(z) \end{aligned} $$Take the integral representation of the left side:
$$ \begin{aligned} &\log G_a(z+1) - \log G_a(z) \notag \\ &= \left( \frac{\partial}{\partial s} \left. \frac{\Gamma(1-s)}{2\pi i} \oint_H \frac{e^{-(z+1)t}(-t)^{s-1}}{(1-e^{-t})^{a}} \, dt \right|_{s=0} - C_a \right) \notag \\ &- \left( \frac{\partial}{\partial s} \left. \frac{\Gamma(1-s)}{2\pi i} \oint_H \frac{e^{-zt}(-t)^{s-1}}{(1-e^{-t})^{a}} \, dt \right|_{s=0} - C_a \right) \\ &= \frac{\partial}{\partial s} \left. \frac{\Gamma(1-s)}{2\pi i} \oint_H \left( \frac{e^{-(z+1)t}(-t)^{s-1}}{(1-e^{-t})^{a}} - \frac{e^{-zt}(-t)^{s-1}}{(1-e^{-t})^{a}} \right) dt \right|_{s=0} \notag \end{aligned} $$Factoring \(e^{(-z+1)t} - e^{-zt} = -e^{-zt} (1-e^{-t})\) and canceling that power from the demoninator yields:
$$ \begin{aligned} & \frac{\partial}{\partial s} \left. \frac{\Gamma(1-s)}{2\pi i} \oint_H \frac{e^{-zt}(-t)^{s-1}}{(1-e^{-t})^{a-1}} \, dt \right|_{s=0} \\ &= \log G_{a-1}(z) \end{aligned} $$Which directly confirms \(\log G_n(z+1) - \log G_n(z) = \log G_{n-1}(z)\), thus guaranteeing the recurrence relationship \(G_n(z+1) = G_{n-1}(z)G_n(z)\). ◻
References #
Abramowitz, M., and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. United States Department of Commerce, 1964.
Nishizawa, M. “Generalized Hölder’s Theorem for Vignéras’ Multiple Gamma Function.” Tokyo Journal of Mathematics, vol. 24, no. 1, 2001, pp. 221-229. https://scispace.com/pdf/generalized-holder-s-theorem-for-vigneras-multiple-gamma-51z4tnq0wp.pdf
Nishizawa, M. “On a q-analogue of the multiple gamma functions.” arXiv preprint q-alg/9505013, 1995. https://arxiv.org/pdf/q-alg/9505013
Ono, K., and A. Singh. “Remarks on MacMahon’s \(q\)-series.” Journal of Combinatorial Theory, Series A, vol. 207, 2024, 105921. https://doi.org/10.1016/j.jcta.2024.105921
Vignéras, M.-F. “L’équation fonctionnelle de la fonction zêta de Selberg du groupe modulaire \(SL(2, \mathbb{Z})\).” Astérisque, vol. 61, 1979, pp. 235-249. https://www.numdam.org/article/AST_1979__61__235_0.pdf
Weisstein, Eric W. “Barnes G-Function.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/BarnesG-Function.html