q-Pi
المؤلف:
Sloane, N. J. A.
المصدر:
Sequences A144874 and A144875 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
28-8-2019
1583
q-Pi
The q-analog of pi 
can be defined by setting 
in the q-factorial
![[a]_q!=1(1+q)(1+q+q^2)...(1+q+...+q^(a-1))](http://mathworld.wolfram.com/images/equations/q-Pi/NumberedEquation1.gif) |
(1)
|
to obtain
![1=sin_q^*(1/2pi)=(pi_q)/(([-1/2]_(q^2)!)^2q^(1/4)),](http://mathworld.wolfram.com/images/equations/q-Pi/NumberedEquation2.gif) |
(2)
|
where 
is Gosper's q-sine, so
(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.
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