q-Exponential Function

The exponential function has two different natural -extensions, denoted  and . They are defined by

			(1)
			(2)
			(3)
			(4)
			(5)

(Koekoek and Swarttouw 1998, p. 18), where  is a q-hypergeometric function.

 has the special form

(6)

for .

The -exponential functions are related to the q-cosine and q-sine by

			(7)
			(8)

REFERENCES:

Koekoek, R. and Swarttouw, R. F. "The q-Gamma Function and the q-Binomial Coefficient." §0.3 in 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, pp. 18-19, 1998.