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

DOI:
https://doi.org/10.1090/S0002-9939-03-07025-4

Published electronically:
November 13, 2003

MathSciNet review:
2045407

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Abstract | References | Similar Articles | Additional Information

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

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