On $q$-analogues of the Euler constant and Lerch’s limit formula
HTML articles powered by AMS MathViewer
- by Nobushige Kurokawa and Masato Wakayama
- Proc. Amer. Math. Soc. 132 (2004), 935-943
- DOI: https://doi.org/10.1090/S0002-9939-03-07025-4
- Published electronically: November 13, 2003
- PDF | Request permission
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 $\gamma (q)$.References
- Richard Askey, The $q$-gamma and $q$-beta functions, Applicable Anal. 8 (1978/79), no. 2, 125–141. MR 523950, DOI 10.1080/00036817808839221
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- Y. Hashimoto, Y. Iijima, N. Kurokawa and M. Wakayama, Euler’s constants for the Selberg and the Dedekind zeta functions, Bulletin of the Belgian Mathematical Society Simon Stevin (to appear).
- F. H. Jackson, On $q$-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193–203.
- M. Kaneko, N. Kurokawa and M. Wakayama, A variation of Euler’s approach to values of the Riemann zeta function, Kyushu Math. J. 57 (2003), 175–192.
- N. Kurokawa and M. Wakayama, A comparison between the sum over Selberg’s zeroes and Riemann’s zeroes, J. Ramanujan Math. Soc. 18 (2003), 221–236. (Errata will also appear.)
- M. Lerch, Dalši studie v oboru Malmsténovských řad, Rozpravy České Akad. 3 No. 28 (1894), 1–61.
- Daniel S. Moak, The $q$-gamma function for $q>1$, Aequationes Math. 20 (1980), no. 2-3, 278–285. MR 577493, DOI 10.1007/BF02190519
- Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Bibliographic 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
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2045407