Göllnitz-Gordon Identities

Author

Eric W. Weisstein
September 7, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Goellnitz-GordonIdentities.nb.

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

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

Modular Identities

First

Goellnitz1[n_]:=Module[{l=Partitions[n]},
{
Select[l,Min[Subtract@@@Subsets[#,{2}]]≥2&&Min[Subtract@@@Subsets[Select[#,EvenQ],{2}]]≥4&],
Select[l,Complement[Mod[#,8],{1,4,7}]==={}&]
}
]

Goellnitz1/@Range[0,7]//ColumnForm

l=Map[Length,Goellnitz1/@Range[0,20],{2}];//Timing

And@@Equal@@@l

First/@l

Second

Goellnitz2[n_]:=Module[{l=Partitions[n]},
{
Select[l,Min[#]>2&&Min[Subtract@@@Subsets[#,{2}]]≥2&&Min[Subtract@@@Subsets[Select[#,EvenQ],{2}]]≥4&],
Select[l,Complement[Mod[#,8],{3,4,5}]==={}&]
}
]

Goellnitz2/@Range[0,11]//ColumnForm

l=Map[Length,Goellnitz2/@Range[0,20],{2}];//Timing

And@@Equal@@@l

First/@l

Sums

(∑_(n=0)^∞)(((q^(n^2)) (QPochhammer[-q, (q^2), n]))/(QPochhammer[(q^2), (q^2), n]))==(1/((QPochhammer2[q, (q^8)]) (QPochhammer2[(q^4), (q^8)]) (QPochhammer2[(q^7), (q^8)])))

Sum[(((q^(n^2))QPochhammer[-q,(q^2),n])/(QPochhammer[(q^2),(q^2),n])),{n,0,∞}]//InputForm

Series[Sum[(((q^(n^2))QPochhammer[-q,(q^2),n])/(QPochhammer[(q^2),(q^2),n])),{n,0,10}],{q,0,20}]

Series[(1/(QPochhammer[q,(q^8)]QPochhammer[(q^4),(q^8)]QPochhammer[(q^7),(q^8)])),{q,0,20}]

CoefficientList[%,q]

(∑_(n=0)^∞)(((q^(n (n+2))) (QPochhammer[-q, (q^2), n]))/(QPochhammer[(q^2), (q^2), n]))==(1/((QPochhammer2[(q^3), (q^8)]) (QPochhammer2[(q^4), (q^8)]) (QPochhammer2[(q^5), (q^8)])))

Sum[(((q^(n(n+2)))QPochhammer[-q,(q^2),n])/(QPochhammer[(q^2),(q^2),n])),{n,0,∞}]//InputForm

Series[Sum[(((q^(n(n+2)))QPochhammer[-q,(q^2),n])/(QPochhammer[(q^2),(q^2),n])),{n,0,10}],{q,0,20}]

Series[(1/(QPochhammer[(q^3),(q^8)]QPochhammer[(q^4),(q^8)]QPochhammer[(q^5),(q^8)])),{q,0,20}]

CoefficientList[%,q]