q-Series Identities

Author

Eric W. Weisstein
September 7, 2004

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

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

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

Adidtional Authors

Identities discovered by 
Michael Trott
December 19, 2000

q-Binomial Theorem

(∑_(n=0)^∞)(((z^n) (QPochhammer[a, q, n]))/(QPochhammer[q, q, n]))==((QPochhammer2[a z, q])/(QPochhammer2[z, q]))

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

Series[Sum[(z^n*QPochhammer[a, q, n])/QPochhammer[q, q, n], {n, 0, 200}]-((QPochhammer[a z,q])/(QPochhammer[z,q]))/.{z->RandomComplex[WorkingPrecision->20],a->RandomComplex[WorkingPrecision->20]},{q,0,10}]//Chop

Euler

(QPochhammer2[a q, q])==(∑_(k=0)^∞)(((((-1))^k) (a^k) (q^((1/2) k (k+1))))/(QPochhammer[q, q, k]))

Sum[(((((-1))^k)(q^(k(k+1)/2))(a^k))/(QPochhammer[q,q,k])),{k,0,∞}]//InputForm

Gordon and McIntosh

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

Sum[((QPochhammer[-q,(q^2),n](q^(n(n-1)))(z^n))/(QPochhammer[z,(q^2),n])),{n,0,∞}]

Sum[((QPochhammer[-z q,(q^4),n](q^(n(2n-1)))(z^n))/(QPochhammer[z,(q^2),2n+1])),{n,0,∞}]

Hirschhorn

(q^3) (mod 5)

PolynomialMod[Normal[Series[QSeries[q,500]^3,{q,0,100}]],5]

CoefficientList[%,q]

PolynomialMod[Sum[(((-1))^n)(2n+1)(q^(n(n+1)/2)),{n,0,13}],5]

X=Product[(1-(q^(25n-15)))(1-(q^(25n-10)))(1-(q^(25n))),{n,5}];
Y=Product[(1-(q^(25n-20)))(1-(q^(25n-5)))(1-(q^(25n))),{n,5}];

Normal[Series[PolynomialMod[X+2q Y,5],{q,0,91}]]

X=Sum[(((-1))^n)(q^((25(n^2)-5n)/2)),{n,-5,5}];
Y=Sum[(((-1))^n)(q^((25(n^2)-15n)/2)),{n,-5,5}];

Normal[Series[PolynomialMod[X+2q Y,5],{q,0,91}]]

(q^5) (mod 5)

PolynomialMod[Normal[Series[QSeries[q,500]^5,{q,0,100}]],5]

PolynomialMod[Normal[Series[QSeries[q^5,500],{q,0,100}]],5]

First/@Partition[CoefficientList[PolynomialMod[Normal[Series[QSeries[q^5,500],{q,0,100}]],5],q],5]

Zucker Notation

(n)==(∏_k^∞)(1-(q^(k n)))==(QPochhammer2[(q^n), (q^n)])

Product[1-(q^(k n)),{k,∞}]

[n]==(q^(n/24))(∏_k^∞)(1-(q^(k n)))==(QPochhammer2[(q^n), (q^n)])

(q^(n/24 ))Product[1-(q^(k n)),{k,∞}]

FullSimplify[%]

Trott's Identities

(n)

From mtrott 2000/12/19.

ids={
-5 qProd[1,1] qProd[2,2] qProd[5,4]+4 qProd[2,5] qProd[5,1] qProd[10,1]+qProd[1,5] qProd[10,2],
-9 qProd[1,1] qProd[2,4] qProd[3,8]+8 qProd[2,9] qProd[3,3] qProd[6,1]+qProd[1,9] qProd[6,4],
-25 qProd[1,2] qProd[2,2] qProd[5,7]+16 qProd[2,8] qProd[5,1] qProd[10,2]+5 qProd[1,6] qProd[5,3] qProd[10,2]+4 qProd[1,5] qProd[2,3] qProd[10,3],
-4 qProd[1,2] qProd[3,2] qProd[4,8]+3 qProd[2,8] qProd[6,4]+qProd[1,8] qProd[4,2] qProd[12,2],
-2 qProd[1,4] qProd[4,14]+qProd[2,14] qProd[8,4]+qProd[1,8] qProd[2,2] qProd[4,4] qProd[8,4]
};

MakeBoxes[qProd[n_,1],TraditionalForm]:=RowBox[{"(",n,")"}]
MakeBoxes[qProd[n_,p_],TraditionalForm]:=SuperscriptBox[RowBox[{"(",n,")"}],p]

ids//ColumnForm//TraditionalForm

The fifth is a Jacobi identity in disguise

Series[ids[[1]]/.n->100,{q,0,200}]

((q^n);(q^k))

Michael Trott
pers. comm.
7/8/2009