Duplication formulae involving Jacobi theta functions and Gosper's -trigonometric functions

Author:
István Mező

Journal:
Proc. Amer. Math. Soc. **141** (2013), 2401-2410

MSC (2010):
Primary 33E05

DOI:
https://doi.org/10.1090/S0002-9939-2013-11576-5

Published electronically:
March 26, 2013

MathSciNet review:
3043021

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using the -trigonometric definitions of Gosper, we devise a new -exponential function. Based on this concept, we derive a number of identities involving the Jacobi theta functions. These considerations lead to the answers to Gosper's ``mysteries''.

**[1]**G. E. Andrews, R. A. Askey and R. Roy,*Special Functions*, Cambridge Univ. Press, Cambridge, 1999. MR**1688958 (2000g:33001)****[2]**J. V. Armitage and W. F. Eberlein,*Elliptic Functions*, Cambridge Univ. Press, Cambridge, 2006. MR**2266844 (2008k:33070)****[3]**G. Gasper and M. Rahman,*Basic Hypergeometric Series*(second edition), Cambridge Univ. Press, Cambridge, 2004. MR**2128719 (2006d:33028)****[4]**R. W. Gosper, Experiments and discoveries in q-trigonometry, in*Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics*. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105. MR**1880081 (2003j:33047)****[5]**R. W. Gosper, Strip mining in the abandoned orefields of nineteenth century mathematics, in*Computers in Mathematics*, Lecture Notes in Pure and Appl. Math., 125, Dekker, New York, 1990, pp. 261-284. MR**1068539 (91h:11154)****[6]**R. W. Gosper, M. E. H. Ismail and R. Zhang, On some strange summation formulas, Illinois J. Math. 37 (1993), 240-277. MR**1208821 (95g:33025)****[7]**M. E. H. Ismail, Y. Takeuchi and R. Zhang, Pages from the computer files of R. William Gosper, Proc. Amer. Math. Soc. 119 (1993), 747-760. MR**1179588 (94h:33001)****[8]**S. Suslov,*An Introduction to Basic Fourier Series*, Kluwer, Dordrecht, Netherlands, 2003. MR**1978912 (2004h:33002)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
33E05

Retrieve articles in all journals with MSC (2010): 33E05

Additional Information

**István Mező**

Affiliation:
Department of Applied Mathematics and Probability Theory, Faculty of Informatics, University of Debrecen, P.O. Box 12, H-4010, Debrecen, Hungary

Email:
mezo.istvan@inf.unideb.hu

DOI:
https://doi.org/10.1090/S0002-9939-2013-11576-5

Keywords:
Jacobi theta functions,
$q$-sine function,
$q$-exponential function,
duplication formulae

Received by editor(s):
February 4, 2011

Received by editor(s) in revised form:
October 17, 2011, and October 20, 2011

Published electronically:
March 26, 2013

Additional Notes:
This research was supported by OTKA grant No. K75566.

Communicated by:
Kathrin Bringmann

Article copyright:
© Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.