Cosine

There are several q-analogs of the cosine function.

The two natural definitions of the -cosine 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 -cosine considered by Gosper (2001) is given by

			(6)
			(7)
			(8)
			(9)

where  is a Jacobi theta function and  is defined via

(10)

This is an even function of unit amplitude, period , and double and triple angle formulas and addition formulas which are analogous to ordinary sine and cosine. For example,

			(11)
			(12)

where  is the q-sine, and  is q-pi (Gosper 2001). The -cosine also satisfies

(13)

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.