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A144874
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Coefficients of the series expansion of q^(-1/4) pi_q.
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1
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1, 2, 0, 0, 1, -2, 1, 2, -3, 0, 2, 0, -1, 0, -1, 0, 4, -2, -2, 0, -1, 4, 1, -4, 0, 2, -2, 0, 2, 0, -1, 2, -1, -4, 2, 0, 2, 2, -2, 0, -2, -2, 3, 2, -3, 0, 4, -2, -2, 2, -2, 2, 0, -4, 0, 4, 3, -2, -1, -2, 0, 2, -2, -2, 2, 2, 2, 0, -4, 0, 2, -2, 1, 2, -3, -2, 4, 0, -2, 2, -2, 4, 0, -4, 2, -2, -2, 2, 2, -2, -1, 4, 1, -2, 2, -2, -4, 2, 0, 0, 2
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OFFSET
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0,2
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COMMENTS
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The gamma function Gamma(x) has a q-extension or q-analog called the q-gamma function, denoted Gamma(q,x), defined by means of the product Gamma(q,x) := 1/(1-q)^(x-1)*( product{n >= 1} (1 - q^n)/(1 - q^(n+x-1)) ) when |q| < 1.
The gamma and q-gamma functions are related through the limiting process Gamma(x) = lim {q -> 1 from below} Gamma(q,x).
It is well known that the constant Pi = Gamma(1/2)^2. This suggests defining a function Pi(q), a q-analog of Pi, by putting Pi(q) = Gamma(q^2,1/2)^2 = (1 - q^2)*( product {n >= 1} (1 - q^(2*n))/(1 - q^(2*n-1)) )^2 = 1 + 2*q + q^4 - 2*q^5 + q^6 + .... This sequence gives the coefficients in the Maclaurin expansion of Pi(q).
Several classical formulas involving Pi have generalizations that involve the function Pi(q). See the Formula section below. (End)
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REFERENCES
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R. Roy, Sources in the development of mathematics, Cambridge University Press 2011.
R. W. Gosper, Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105.
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LINKS
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Eric Weisstein's World of Mathematics, q-Pi
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FORMULA
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Pi(q) = q^(1/4)*pi_q.
Pi(q) = (1 - q^2)*( Sum_{n >=0} q^(n*(n+1)/2) )^2.
Some q-analogs of classical formulas
= = = = = = = = = = = = = = = = = = =
Let [n] := 1 + q + q^2 + ... + q^(n-1) denote the q-analog of the natural number n.
(a) Wallis' formula Pi/2 = (2/1)*(2/3)*(4/3)*(4/5)*(6/5)*(6/7)* ....
q_analog: Pi(q)/[2] = ([2]/[1])*([2]/[3])*([4]/[3])*([4]/[5])*([6]/[5])*([6]/[7)* ....
(b) The Euler-Sylvester continued fraction Pi/2 = 1 + 1/(1 + 2/(1 + 6/(1 + 12/(1 + ...)))) (Roy 3.47 and 3.67).
q-analog: Pi(q)/[2] = 1 + q/(1 + q*[1]*[2]/(1 + q*[2]*[3]/(1 + q*[3]*[4]/(1 + ...)))).
(c) The Madhava-Leibniz series Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ....
We have two q-analogs:
Pi(q^2)/[4] = 1/[1] - q/[3] + q^2/[5] - q^3/[7] + ...,
as well as
Pi(q)/[2] = sum {n in Z} (-1)^n*q^(n*(n+1))/[2*n+1].
(d) The result Pi^2/8 = sum {n >= 0} 1/(2*n+1)^2.
q-analog: Pi(q^2)^2/[2]^2 = (1 + q)/[1]^2 + q*(1 + q^3)/[3]^2 + q^2*(1 + q^5)/[5]^2 + ....
(e) The result Pi^4/96 = sum {n >= 0} 1/(2*n+1)^4.
q-analog: q*Pi(q^2)^4/[2]^4 = f(q)/[1]^4 + f(q^3)/[3]^4 + f(q^5)/[5]^4 + ..., where f(q) = q + 4*q^2 + q^3. (End)
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EXAMPLE
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G.f. = 1 + 2*x + x^4 - 2*x^5 + x^6 + 2*x^7 - 3*x^8 + 2*x^10 + ...
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MATHEMATICA
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max = 100; pi[q_] := (1 - q^2)*q^(1/4)*Product[(1 - q^(2n))^2 / (1 - q^(2n - 1))^2, {n, 1, max}]; CoefficientList[ Series[ q^(-1/4)*pi[q], {q, 0, max}], q] (* Jean-François Alcover, Feb 07 2013 *)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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