Göllnitz's Theorem

Author

Eric W. Weisstein
September 7, 2008

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

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

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

Products

(∏_(q=0)^∞)((q^(6 n+2))+1) ((q^(6 n+4))+1) ((q^(6 n+5))+1)==(QPochhammer2[-(q^2), (q^6)]) (QPochhammer2[-(q^4), (q^6)]) (QPochhammer2[-(q^5), (q^6)])

expr1=Product[(1+(q^(6n+2)))(1+(q^(6n+4)))(1+(q^(6n+5))),{n,0,∞}]

Series[expr1,{q,0,20}]//Normal

(∏_(n=0)^∞)(((1-(q^(12 n+4))) (1-(q^(12 n+8))) (1-(q^(12 n+10))))/((1-(q^(6 n+2))) (1-(q^(6 n+4))) (1-(q^(6 n+5)))))==(((QPochhammer2[(q^4), (q^12)]) (QPochhammer2[(q^8), (q^12)]) (QPochhammer2[(q^10), (q^12)]))/((QPochhammer2[(q^2), (q^6)]) (QPochhammer2[(q^4), (q^6)]) (QPochhammer2[(q^5), (q^6)])))

expr2=Product[(((1-(q^(12n+4)))(1-(q^(12n+8)))(1-(q^(12n+10))))/((1-(q^(6n+2)))(1-(q^(6n+4)))(1-(q^(6n+5))))),{n,0,∞}]

Series[expr2,{q,0,20}]//Normal

(∏_(n=0)^∞)(1/((1-(q^(12 n+2))) (1-(q^(12 n+5))) (1-(q^(12 n+11)))))==(1/((QPochhammer2[(q^2), (q^12)]) (QPochhammer2[(q^5), (q^12)]) (QPochhammer2[(q^11), (q^12)])))

expr3=Product[(1/((1-(q^(12n+2)))(1-(q^(12n+5)))(1-(q^(12n+11))))),{n,0,∞}]

Series[expr3,{q,0,20}]//Normal

CoefficientList[1+(q^2)+(q^4)+(q^5)+(q^6)+(q^7)+(q^8)+(q^9)+2 (q^10)+2 (q^11)+2 (q^12)+2 (q^13)+3 (q^14)+3 (q^15)+4 (q^16)+4 (q^17)+4 (q^18)+5 (q^19)+5 (q^20),q]

Identities

{expr1,expr2,expr3}

FullSimplify[Subtract@@@Subsets[{expr1,expr2,expr3},{2}]]

Series[%,{q,0,100}]//Normal

Identities

partitions[n_]:=partitions[n]=Partitions[n];

a[n_]:=a[n]=Select[partitions[n],And@@Function[m,Function[c,Mod[m,12]==c]/@Or@@{2,5,11}]/@#&]

b[n_]:=b[n]=Select[partitions[n],Max[Length/@Split[#]]==1&&And@@Function[m,Function[c,Mod[m,6]==c]/@Or@@{2,4,5}]/@#&]

c[n_]:=c[n]=Module[{p=Partition[#,2,1]&/@Select[partitions[n],Last[#]≠1&&Last[#]≠3&],cc},
cc=Append[First/@#,#[[-1,-1]]]&/@(DeleteCases[Select[p,And@@Function[f,Subtract@@f>If[Function[c,Mod[First[f],6]==c]/@Or@@{0,1,3},6,5]]/@#&],{}]);
If[n==1||n==3,cc,Prepend[cc,{n}]]
]

Table[Length/@{a[n],b[n],c[n]},{n,20}]

Table[Length[a[n]],{n,30}]

Table[Length[b[n]],{n,30}]

Table[Length[c[n]],{n,30}]

CoefficientList[Product[(1+q^(6n+2))(1+q^(6n+4))(1+q^(6n+5)),{n,0,3}],q]

Take[%84,3 6]

Take[CoefficientList[Product[(1+q^(6n+2))(1+q^(6n+4))(1+q^(6n+5)),{n,0,40}],q],
100]