Duplication formulae involving Jacobi theta functions and Gosper's trigonometric functions
Author:
István Mező
Journal:
Proc. Amer. Math. Soc. 141 (2013), 24012410
MSC (2010):
Primary 33E05
Published electronically:
March 26, 2013
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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''.
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J. V. Armitage and W. F. Eberlein, Elliptic Functions, Cambridge Univ. Press, Cambridge, 2006. MR 2266844 (2008k:33070)
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G. Gasper and M. Rahman, Basic Hypergeometric Series (second edition), Cambridge Univ. Press, Cambridge, 2004. MR 2128719 (2006d:33028)
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R. W. Gosper, Experiments and discoveries in qtrigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79105. MR 1880081 (2003j:33047)
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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. 261284. MR 1068539 (91h:11154)
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R. W. Gosper, M. E. H. Ismail and R. Zhang, On some strange summation formulas, Illinois J. Math. 37 (1993), 240277. MR 1208821 (95g:33025)
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M. E. H. Ismail, Y. Takeuchi and R. Zhang, Pages from the computer files of R. William Gosper, Proc. Amer. Math. Soc. 119 (1993), 747760. MR 1179588 (94h:33001)
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S. Suslov, An Introduction to Basic Fourier Series, Kluwer, Dordrecht, Netherlands, 2003. MR 1978912 (2004h:33002)
 [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 qtrigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79105. 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. 261284. 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), 240277. 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), 747760. MR 1179588 (94h:33001)
 [8]
 S. Suslov, An Introduction to Basic Fourier Series, Kluwer, Dordrecht, Netherlands, 2003. MR 1978912 (2004h:33002)
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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, H4010, Debrecen, Hungary
Email:
mezo.istvan@inf.unideb.hu
DOI:
http://dx.doi.org/10.1090/S000299392013115765
PII:
S 00029939(2013)115765
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.
