Transcendence of the log gamma function and some discrete periods

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Abstract

We study transcendental values of the logarithm of the gamma function. For instance, we show that for any rational number x with 0 < x < 1, the number log Γ (x) + log Γ (1 - x) is transcendental with at most one possible exception. Assuming Schanuel's conjecture, this possible exception can be ruled out. Further, we derive a variety of results on the Γ-function as well as the transcendence of certain series of the form ∑n = 1∞ P (n) / Q (n), where P (x) and Q (x) are polynomials with algebraic coefficients. © 2009 Elsevier Inc. All rights reserved.

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Gun, S., Murty, M. R., & Rath, P. (2009). Transcendence of the log gamma function and some discrete periods. Journal of Number Theory, 129(9), 2154–2165. https://doi.org/10.1016/j.jnt.2009.01.008

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