q-Pi

Author

Eric W. Weisstein
October 2, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/q-Pi.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/q-Pi.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Initialization

QPi[q_]:=((QFactorial[-(1/2),(q^2)])^2)(q^(1/4))

Notation

QPi[q]//TraditionalForm

((QFactorial[-(1/2),(q^2)])^2)//TraditionalForm

Definitions

QPi/@{.2,.3}

(1/2) (1-(q^2)) (EllipticTheta[2, 0, q]) (EllipticTheta[3, 0, q])

QPi2[q_]:=(1/2)EllipticTheta[2,0,q]EllipticTheta[3,0,q](1-(q^2))

QPi2/@{.2,.3}

FullSimplify[QPi2[q]]

FullSimplify[QPi[q]-QPi2[q]]

(1/4) (1-(q^2)) ((EllipticTheta[2, 0, (Sqrt[q])])^2)

QPi3[q_]:=(1/4)((EllipticTheta[2,0,(Sqrt[q])])^2)(1-(q^2))

QPi3/@{.2,.3}

FullSimplify[QPi3[q]]

FullSimplify[QPi3[q]-QPi[q]]

(1-(q^2))(q^(1/4)) (∏_n^∞)((((1-(q^(2 n))))^2)/(((1-(q^(2 n-1))))^2))

(q^(1/4))Product[((((1-(q^(2n))))^2)/(((1-(q^(2n-1))))^2)),{n,∞}]//FullSimplify

((1-(q^2))(q^(1/4))Product[((((1-(q^(2n))))^2)/(((1-(q^(2n-1))))^2)),{n,60}]/.q->#)&/@{.2,.3}

Product[(((1-(q^(2n))))^2),{n,∞}]

Product[(((1-(q^(2n-1))))^2),{n,∞}]

!=(((∑_(n=-∞)^∞)(4 n+1) (q^(2 n+(1/2))))/((∑_(n=-∞)^∞)(((-q))^(n^2))))

((Sum[(4n+1)(q^(2n+1/2)),{n,-∞,∞}])/(Sum[(((-q))^(n^2)),{n,-∞,∞}]))

Sum[(4n+1)(q^(2n+1/2)),{n,-∞,∞}]

Sum[(4n+1)(q^(2n+1/2)),{n,-10,10}]/.q->.2

Sum[(((-q))^(n^2)),{n,-∞,∞}]

Plot

Plot[QPi[q],{q,0,.99},PlotStyle->Red,PlotRange->All,AxesLabel->{q,(Subscript[π, q])}]

Series

Series[(q^(-1/4))QPi[q],{q,0,15}]

CoefficientList[%,q]

FindSequenceFunction[%]

Series[(q^(-1/4))QPi2[q],{q,0,15}]

SeriesCoefficient[(q^(-1/4))QPi2[q],{q,0,n}]

Series[(q^(-1/4))QPi3[q],{q,0,15}]

SeriesCoefficient[(q^(-1/4))QPi3[q],{q,0,n}]

Derivative and Integral

Derivatives

D[QPi[q],q]

FullSimplify[%]

((QFactorial[-(1/2),(q^2)] (QFactorial[-(1/2),(q^2)]+16 (q^2) Derivative[0,1][QFactorial][-(1/2),(q^2)]))/(4 (q^(3/4))))/.q->.6

D[QPi2[q],q]//FullSimplify

D[QPi3[q],q]//FullSimplify

Plot[((QFactorial[-(1/2),(q^2)] (QFactorial[-(1/2),(q^2)]+16 (q^2) Derivative[0,1][QFactorial][-(1/2),(q^2)]))/(4 (q^(3/4)))),{q,.1,.9},WorkingPrecision->5,PlotRange->All]

With[{h=10^-7},Plot[((QPi[q+h]-QPi[q])/h),{q,0,.95}]]

Second Derivative

With[{h=10^-10},Plot[((QPi[q+2h]-2QPi[q+h]+QPi[q])/(h^2)),{q,.1,.95},WorkingPrecision->30]]

With[{h=10^-10},Plot[((QPi[q+2h]-2QPi[q+h]+QPi[q])/(h^2)),{q,.2,.3},WorkingPrecision->30]]

With[{h=10^-30},FindRoot[((QPi[q+2h]-2QPi[q+h]+QPi[q])/(h^2))==0,{q,2/10,25/100},WorkingPrecision->100]]

With[{h=10^-50},FindRoot[((QPi[q+2h]-2QPi[q+h]+QPi[q])/(h^2))==0,{q,2/10,25/100},WorkingPrecision->500]]

D[QPi2[q],{q,2}]

D[QPi3[q],{q,2}]

Integral

Integrate[QPi[q],q]

Integrate[QPi2[q],q]

Integrate[QPi3[q],q]

NIntegrate[QPi[q],{q,0,1},WorkingPrecision->50]

Integrate[QPi[q],{q,0,1}]

Integrate[QPi2[q],{q,0,1}]

NIntegrate[QPi2[q],{q,0,1},WorkingPrecision->50]

Integrate[QPi3[q],{q,0,1}]

NIntegrate[QPi3[q],{q,0,1},WorkingPrecision->50]

Special Values

QPi/@{0,1/GoldenRatio,1/4,1/2,1}

N[%,40]

FunctionExpand

FunctionExpand[QPi[q]]

Gosper Relation

id={((4QPi[(q^4)])/((q^4)+1)),((((((q^2)+1))^2)((QPi[q])^2))/(QPi[(q^2)]))-((((q^4)+1)((QPi[(q^2)])^2))/(QPi[(q^4)]))}

FullSimplify[Subtract@@id,0<q<1]

FunctionExpand[%]

Table[Subtract@@id/.q->RandomReal[],{10^3}]//Chop//Union

Limits

Limit[QPi[q],q->1]

QPi[q]/.q->1.

Limit[QPi2[q],q->1,Direction->1]

QPi2[.999999]

QPi2[1.]

Limit[QPi3[q],q->1,Direction->1]

QPi3[.999999]

QPi3[1.]