The qanalog of pi can be defined by setting in the qfactorial

(1)

to obtain

(2)

where is Gosper's qsine,
so
(Gosper 2001).
It has the Maclaurin series

(7)

(OEIS A144874).
It is related to the qanalog 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)

See also
Pi,
qAnalog,
qCosine,
qExponential
Function,
qFactorial,
qSine,
Wallis Formula
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References
Sloane, N. J. A. Sequences A144874 and A144875 in "The OnLine Encyclopedia
of Integer Sequences."Gosper, R. W. "Experiments and
Discoveries in qTrigonometry." In Symbolic
Computation, Number Theory,Special Functions, Physics and Combinatorics. Proceedings
of the Conference Held at the University of Florida, Gainesville, FL, November 1113,
1999 (Ed. F. G. Garvan and M. E. H. Ismail). Dordrecht,
Netherlands: Kluwer, pp. 79105, 2001.Referenced on WolframAlpha
qPi
Cite this as:
Weisstein, Eric W. "qPi." From MathWorldA
Wolfram Web Resource. https://mathworld.wolfram.com/qPi.html
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