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

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

Offset: 0

Author

Eric W. Weisstein, Sep 23 2008

Keywords

• sign

Comments

From Peter Bala, Dec 12 2013: (Start)

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)

Examples

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

References

• 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.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• R. W. Gosper, Experiments and discoveries in q-trigonometry, Preprint.
• R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts
• K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94
• Eric Weisstein's World of Mathematics, q-Pi
• Wikipedia, Leibniz formula for Pi
• Wikipedia, q-analog
• Wikipedia, q-gamma function
• Wikipedia, Wallis product

Crossrefs

Cf. A008441.

Programs

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] (* Jean-François Alcover, Feb 07 2013 *)

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 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)

a(n) = A008441(n) - A008441(n-2) for n > 1. - Seiichi Manyama, Jan 05 2022