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

			(1)
			(2)
			(3)

where  and  are q-exponential functions. The -cosine and -sine functions satisfy the relations

			(4)
			(5)

Another definition of the -sine considered by Gosper (2001) is given by

			(6)
			(7)
			(8)

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.