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

Published electronically:
March 26, 2013

MathSciNet review:
3043021

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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]**George E. Andrews, Richard Askey, and Ranjan Roy,*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958****[2]**J. V. Armitage and W. F. Eberlein,*Elliptic functions*, London Mathematical Society Student Texts, vol. 67, Cambridge University Press, Cambridge, 2006. MR**2266844****[3]**George Gasper and Mizan Rahman,*Basic hypergeometric series*, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR**2128719****[4]**R. Wm. Gosper,*Experiments and discoveries in 𝑞-trigonometry*, Symbolic computation, number theory, special functions, physics and combinatorics (Gainesville, FL, 1999) Dev. Math., vol. 4, Kluwer Acad. Publ., Dordrecht, 2001, pp. 79–105. MR**1880081**, 10.1007/978-1-4613-0257-5_6**[5]**William Gosper,*Strip mining in the abandoned orefields of nineteenth century mathematics*, Computers in mathematics (Stanford, CA, 1986) Lecture Notes in Pure and Appl. Math., vol. 125, Dekker, New York, 1990, pp. 261–284. MR**1068539****[6]**R. William Gosper, Mourad E. H. Ismail, and Ruiming Zhang,*On some strange summation formulas*, Illinois J. Math.**37**(1993), no. 2, 240–277. MR**1208821****[7]**Mourad E. H. Ismail, Yu Takeuchi, and Ruiming Zhang,*Pages from the computer files of R. William Gosper*, Proc. Amer. Math. Soc.**119**(1993), no. 3, 747–760. MR**1179588**, 10.1090/S0002-9939-1993-1179588-8**[8]**Sergei K. Suslov,*An introduction to basic Fourier series*, Developments in Mathematics, vol. 9, Kluwer Academic Publishers, Dordrecht, 2003. With a foreword by Mizan Rahman. MR**1978912**

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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.