

A144874


Coefficients of the series expansion of q^(1/4) pi_q.


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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

0,2


COMMENTS

From Peter Bala, Dec 12 2013: (Start)
The gamma function Gamma(x) has a qextension or qanalog called the qgamma function, denoted Gamma(q,x), defined by means of the product Gamma(q,x) := 1/(1q)^(x1)*( product{n >= 1} (1  q^n)/(1  q^(n+x1)) ) when q < 1.
The gamma and qgamma 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 qanalog 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*n1)) )^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)


REFERENCES

R. Roy, Sources in the development of mathematics, Cambridge University Press 2011
R. W. Gosper, Experiments and discoveries in qtrigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79105.


LINKS

Table of n, a(n) for n=0..100.
K. Ono, S. Robins, P. T. Wahl,On the representation of integers as sums of triangular numbers, aequationes mathematicae, August 1995, Volume 50, Issue 12, pp 7394
Eric Weisstein's World of Mathematics, qPi
Wikipedia, Leibniz formula for Pi
Wikipedia, qanalog
Wikipedia, qgamma function
Wikipedia, Wallis product


FORMULA

From Peter Bala, Dec 12 2013: (Start)
Pi(q) = q^(1/4)*pi_q.
Pi(q) = (1  q^2)*( sum {n >=0} q^(n*(n+1)/2) )^2.
Some qanalogs of classical formulas
= = = = = = = = = = = = = = = = = = =
Let [n] := 1 + q + q^2 + ... + q^(n1) denote the qanalog 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 EulerSylvester continued fraction Pi/2 = 1 + 1/(1 + 2/(1 + 6/(1 + 12/(1 + ...)))) (Roy 3.47 and 3.67).
qanalog: Pi(q)/[2] = 1 + q/(1 + q*[1]*[2]/(1 + q*[2]*[3]/(1 + q*[3]*[4]/(1 + ...)))).
(c) The MadhavaLeibniz series Pi/4 = 1  1/3 + 1/5  1/7 + ....
We have two qanalogs:
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.
qanalog: 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.
qanalog: 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)


EXAMPLE

G.f. = 1 + 2*x + x^4  2*x^5 + x^6 + 2*x^7  3*x^8 + 2*x^10 + ...


MATHEMATICA

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] (* JeanFrançois Alcover, Feb 07 2013 *)


CROSSREFS

Sequence in context: A204425 A245187 A079113 * A257900 A039971 A205593
Adjacent sequences: A144871 A144872 A144873 * A144875 A144876 A144877


KEYWORD

sign


AUTHOR

Eric W. Weisstein, Sep 23 2008


STATUS

approved



