Dyson mod 27 Identities

Author

Eric W. Weisstein
March 11, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/DysonMod27Identities.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DysonMod27Identities.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Additional Authors

Contributed by
Andrew Sills
Rutgers University
March 11, 2005

Identities

A(q)=1+(∑_(n=1)^∞)((Subscript[(q^(n^2))((q^3);(q^3)), n- 1])/((Subscript[(q;q), n])(Subscript[(q;q), 2n- 1])))=((Subscript[((q^12),(q^15),(q^27);(q^27)), ∞])/(Subscript[(q;q), ∞]))

Series[1+Sum[(((q^(n^2))QSeries[{(q^3),(q^3)},n-1])/(QSeries[{q,q},n]QSeries[{q,q},2n-1])),{n,1,20}],{q,0,20}]

Series[((QSeries[{{(q^12),(q^15),(q^27)},(q^27)}])/(QSeries[{q,q}])),{q,0,20}]

CoefficientList[%,q]

B(q)=(∑_(n=0)^∞)((Subscript[(q^((n^2)+n))((q^3);(q^3)), n])/((Subscript[(q;q), n])(Subscript[(q;q), 2n+1])))=((Subscript[((q^9);(q^9)), ∞])/(Subscript[(q;q), ∞]))

Series[Sum[(((q^((n^2)+n))QSeries[{(q^3),(q^3)},n] )/(QSeries[{q,q},n]QSeries[{q,q},2n+1])),{n,0,20}],{q,0,20}]

Series[((QSeries[{{(q^9)},(q^9)}])/(QSeries[{q,q}])),{q,0,20}]

CoefficientList[%,q]

C(q)=(∑_(n=0)^∞)((Subscript[(q^((n^2)+2n))((q^3);(q^3)), n])/((Subscript[(q;q), n])(Subscript[(q;q), 2n+2])))=((Subscript[((q^6),(q^21),(q^27);(q^27)), ∞])/(Subscript[(q;q), ∞]))

Series[Sum[(((q^((n^2)+2n))QSeries[{(q^3),(q^3)},n] )/(QSeries[{q,q},n]QSeries[{q,q},2n+2])),{n,0,20}],{q,0,20}]

Series[((QSeries[{{(q^6),(q^21),(q^27)},(q^27)}])/(QSeries[{q,q}])),{q,0,20}]

CoefficientList[%,q]

D(q)=(∑_(n=0)^∞)((Subscript[(q^((n^2)+3n))((q^3);(q^3)), n])/((Subscript[(q;q), n])(Subscript[(q;q), 2n+2])))=((Subscript[((q^3),(q^24),(q^27);(q^27)), ∞])/(Subscript[(q;q), ∞]))

Series[Sum[(((q^((n^2)+3n))QSeries[{(q^3),(q^3)},n] )/(QSeries[{q,q},n]QSeries[{q,q},2n+2])),{n,0,20}],{q,0,20}]

Series[((QSeries[{{(q^3),(q^24),(q^27)},(q^27)}])/(QSeries[{q,q}])),{q,0,20}]

CoefficientList[%,q]