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A144874 Coefficients of the series expansion of q^(-1/4) pi_q.

%I A144874 #41 Apr 09 2025 04:21:17
%S A144874 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,
%T A144874 -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,
%U A144874 -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
%N A144874 Coefficients of the series expansion of q^(-1/4) pi_q.
%C A144874 From _Peter Bala_, Dec 12 2013: (Start)
%C A144874 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.
%C A144874 The gamma and q-gamma functions are related through the limiting process Gamma(x) = lim {q -> 1 from below} Gamma(q,x).
%C A144874 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).
%C A144874 Several classical formulas involving Pi have generalizations that involve the function Pi(q). See the Formula section below. (End)
%D A144874 R. Roy, Sources in the development of mathematics, Cambridge University Press 2011.
%D A144874 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.
%H A144874 Seiichi Manyama, <a href="/A144874/b144874.txt">Table of n, a(n) for n = 0..10000</a>
%H A144874 R. W. Gosper, <a href="/A274621/a274621.pdf">Experiments and discoveries in q-trigonometry</a>, Preprint.
%H A144874 R. W. Gosper, <a href="/A274621/a274621_1.pdf">q-Trigonometry: Some Prefatory Afterthoughts</a>
%H A144874 K. Ono, S. Robins and P. T. Wahl, <a href="https://web.archive.org/web/20190712123047/http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/006.pdf">On the representation of integers as sums of triangular numbers</a>, aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94
%H A144874 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-Pi.html">q-Pi</a>
%H A144874 Wikipedia, <a href="http://en.wikipedia.org/wiki/Leibniz_formula_for_pi">Leibniz formula for Pi</a>
%H A144874 Wikipedia, <a href="http://en.wikipedia.org/wiki/Q-analog">q-analog</a>
%H A144874 Wikipedia, <a href="http://en.wikipedia.org/wiki/Q-gamma_function">q-gamma function</a>
%H A144874 Wikipedia, <a href="http://en.wikipedia.org/wiki/Wallis_product">Wallis product</a>
%F A144874 From _Peter Bala_, Dec 12 2013: (Start)
%F A144874 Pi(q) = q^(1/4)*pi_q.
%F A144874 Pi(q) = (1 - q^2)*( Sum_{n >=0} q^(n*(n+1)/2) )^2.
%F A144874 Some q-analogs of classical formulas
%F A144874 = = = = = = = = = = = = = = = = = = =
%F A144874 Let [n] := 1 + q + q^2 + ... + q^(n-1) denote the q-analog of the natural number n.
%F A144874 (a) Wallis' formula Pi/2 = (2/1)*(2/3)*(4/3)*(4/5)*(6/5)*(6/7)* ....
%F A144874 q_analog: Pi(q)/[2] = ([2]/[1])*([2]/[3])*([4]/[3])*([4]/[5])*([6]/[5])*([6]/[7])* ....
%F A144874 (b) The Euler-Sylvester continued fraction Pi/2 = 1 + 1/(1 + 2/(1 + 6/(1 + 12/(1 + ...)))) (Roy 3.47 and 3.67).
%F A144874 q-analog: Pi(q)/[2] = 1 + q/(1 + q*[1]*[2]/(1 + q*[2]*[3]/(1 + q*[3]*[4]/(1 + ...)))).
%F A144874 (c) The Madhava-Leibniz series Pi/4 = 1 - 1/3 + 1/5 - 1/7 + ....
%F A144874 We have two q-analogs:
%F A144874 Pi(q^2)/[4] = 1/[1] - q/[3] + q^2/[5] - q^3/[7] + ...,
%F A144874 as well as
%F A144874 Pi(q)/[2] = sum {n in Z} (-1)^n*q^(n*(n+1))/[2*n+1].
%F A144874 (d) The result Pi^2/8 = sum {n >= 0} 1/(2*n+1)^2.
%F A144874 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 + ....
%F A144874 (e) The result Pi^4/96 = sum {n >= 0} 1/(2*n+1)^4.
%F A144874 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)
%F A144874 a(n) = A008441(n) - A008441(n-2) for n > 1. - _Seiichi Manyama_, Jan 05 2022
%e A144874 G.f. = 1 + 2*x + x^4 - 2*x^5 + x^6 + 2*x^7 - 3*x^8 + 2*x^10 + ...
%t A144874 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 *)
%Y A144874 Cf. A008441.
%K A144874 sign
%O A144874 0,2
%A A144874 _Eric W. Weisstein_, Sep 23 2008