The q-analog of pi  can be defined by setting  in the q-factorial

(1)

to obtain

(2)

where  is Gosper's q-sine, so

			(3)
			(4)
			(5)
			(6)

(Gosper 2001).

It has the Maclaurin series

(7)

(OEIS A144874).

It is related to the q-analog of the Wallis formula (Gosper 2001), and has the special value

(8)

The area under  is given by

(9)

(OEIS A144875).

Gosper has developed an iterative algorithm for computing  based on the algebraic recurrence relation

(10)

REFERENCES:

Sloane, N. J. A. Sequences A144874 and A144875 in "The On-Line Encyclopedia of Integer Sequences."

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.