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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On $q$-analogues of the Euler constant and Lerch's limit formula


Authors: Nobushige Kurokawa and Masato Wakayama
Journal: Proc. Amer. Math. Soc. 132 (2004), 935-943
MSC (2000): Primary 11M35, 33D05
Published electronically: November 13, 2003
MathSciNet review: 2045407
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Abstract: We introduce and study a $q$-analogue $\gamma (q)$ of the Euler constant via a suitably defined $q$-analogue of the Riemann zeta function. We show, in particular, that the value $\gamma (2)$ is irrational. We also present a $q$-analogue of the Hurwitz zeta function and establish an analogue of the limit formula of Lerch in 1894 for the gamma function. This limit formula can be regarded as a natural generalization of the formula of $\gamma (q)$.


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Additional Information

Nobushige Kurokawa
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Meguro, Tokyo, 152-0033 Japan
Email: kurokawa@math.titech.ac.jp

Masato Wakayama
Affiliation: Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka, 812-8581 Japan
Email: wakayama@math.kyushu-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07025-4
PII: S 0002-9939(03)07025-4
Keywords: Euler's constant, Riemann's zeta function, Hurwitz's zeta function, $q$-gamma function, Lerch's limit formula
Received by editor(s): September 3, 2002
Published electronically: November 13, 2003
Additional Notes: Work in part supported by Grant-in-Aid for Scientific Research (B) No. 11440010, and by Grant-in-Aid for Exploratory Research No. 13874004, Japan Society for the Promotion of Science
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2003 American Mathematical Society