On the Barnes double gamma function

08/29/2022

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by   Shahen Alexanian, et al.

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We aim to achieve the following three goals. First of all, we collect all known definitions, transformation properties and functional identities of Barnes double gamma function G(z;τ). Second, we derive an algorithm for numerically computing the double gamma function and present its complete asymptotic expansion as z→∞. Third, we derive some new properties, including a new product identity and new representations of the gamma modular forms C(τ) and D(τ).