q-Sine
المؤلف:
Koekoek, R. and Swarttouw, R. F.
المصدر:
The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17
الجزء والصفحة:
...
31-8-2019
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q-Sine
There are several q-analogs of the sine function.
The two natural definitions of the 
-sine defined by Koekoek and Swarttouw (1998) are given by
where 
and 
are q-exponential functions. The 
-cosine and 
-sine functions satisfy the relations
Another definition of the 
-sine considered by Gosper (2001) is given by
where 
is a Jacobi theta function and 
is defined via
 |
(9)
|
This is an odd function of unit amplitude and period 
with double and triple angle formulas and addition formulas which are analogous to ordinary sine and cosine. For example,
 |
(10)
|
where 
is the q-cosine and 
is q-pi (Gosper 2001).
REFERENCES:
Gosper, R. W. "Experiments and Discoveries in q-Trigonometry." In Symbolic Computation, Number Theory,Special Functions, Physics and Combinatorics. Proceedings of the Conference Held at the University of Florida, Gainesville, FL, November 11-13, 1999 (Ed. F. G. Garvan and M. E. H. Ismail). Dordrecht, Netherlands: Kluwer, pp. 79-105, 2001.
Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its 
-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 18-19, 1998.
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