q-Factorial

The q-analog of the factorial (by analogy with the q-gamma function). For  an integer, the -factorial is defined by

			(1)
			(2)
			(3)

(Koepf 1998, p. 26). For ,

(4)

where  is the q-gamma function.

-factorials are implemented in the Wolfram Language as QFactorial[n, q].

The first few values are

			(5)
			(6)
			(7)
			(8)
			(9)
			(10)

REFERENCES:

Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.

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.

Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 26 and 30, 1998.