login
This site is supported by donations to The OEIS Foundation.

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A144874 Coefficients of the series expansion of q^(-1/4) pi_q. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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)

REFERENCES

R. Roy, Sources in the development of mathematics, Cambridge University Press 2011

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 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

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)

EXAMPLE

q^(-1/4)*(1 + 2*q + q^4 - 2*q^5 + q^6 + 2*q^7 - 3*q^8 + 2*q^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] (* Jean-Fran├žois Alcover, Feb 07 2013 *)

CROSSREFS

Sequence in context: A204425 A245187 A079113 * A039971 A205593 A112020

Adjacent sequences:  A144871 A144872 A144873 * A144875 A144876 A144877

KEYWORD

sign

AUTHOR

Eric W. Weisstein, Sep 23, 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified August 27 22:12 EDT 2014. Contains 246149 sequences.