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q-Factorial

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

[k]_q!=faq(k,q)
(1)
=1(1+q)(1+q+q^2)...(1+q+...+q^(k-1))
(2)
=((q;q)_k)/((1-q)^k)
(3)

(Koepf 1998, p. 26). For k in N,

[k]_q!=Gamma_q(k+1),
(4)

where Gamma_q(k+1) is the q-gamma function.

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

The first few values are

[1]_q!=1
(5)
[2]_q!=1+q
(6)
[3]_q!=(1+q)(1+q+q^2)
(7)
=1+2q+2q^2+q^3
(8)
[4]_q!=(1+q)(1+q+q^2)(1+q+q^2+q^3)
(9)
=1+3q+5q^2+6q^3+5q^4+3q^5+q^6.
(10)

See also

q-Beta Function, q-Binomial Coefficient, q-Bracket, q-Cosine, q-Gamma Function, q-Pi, q-Sine

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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.

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q-Factorial

Cite this as:

Weisstein, Eric W. "q-Factorial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/q-Factorial.html

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