Abstract
We introduce a q-generalization of the sine and cosine functions, related to the ϑ functions, but (as revealed by computer experiments) possessing addition and multiplication formulas more analogous to those of ordinary sin and cos. These formulas then contribute identities to ϑ theory, and hint of a more natural formulation of ϑ functions as outgrowths of elementary functions. Nevertheless, this paper can be read without knowledge of ϑ functions—it was certainly written that way.
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References
F. H. Jackson, A basic-sine and cosine with symbolical solutions of certain differential equations, Proc. Edinburgh Math. Soc. 22 (1904), 28–39.
F. H. Jackson, The basic gamma-function and the elliptic functions, Proc. Roy. Soc. London (A) 76 (1905), 127–144.
F. H. Jackson, Pseudo-periodic functions analogous to the circular functions, Messen-ger of Mathematics 34 (1905), 32–39.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. Cambridge Uni-versity Press, Cambridge, 4th edition, 1927.
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© 2001 Kluwer Academic Publishers
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Gosper, R.W. (2001). Experiments and Discoveries in q-Trigonometry. In: Garvan, F.G., Ismail, M.E.H. (eds) Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, vol 4. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0257-5_6
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DOI: https://doi.org/10.1007/978-1-4613-0257-5_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4020-0101-7
Online ISBN: 978-1-4613-0257-5
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