Jackson-Slater Identity

Author

Eric W. Weisstein
September 21, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Jackson-SlaterIdentity.nb.

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

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

Expansions

(∑_(k=0)^∞)((q^(2 (k^2)))/(QPochhammer[q, q, 2 k]))==(((QPochhammer2[q, (q^8)]) (QPochhammer2[(q^6), (q^16)]) (QPochhammer2[(q^7), (q^8)]) (QPochhammer2[(q^8), (q^8)]) (QPochhammer2[(q^10), (q^16)]))/(QPochhammer2[q, q]))

CoefficientList[Series[Sum[((q^(2(k^2)))/(QSeries[{q,q},2k])),{k,0,50}],{q,0,20}],q]

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

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

Series[(1/(QPochhammer[q]))Product[(1-(q^(8k+1)))(1-(q^(8k+7)))(1-(q^(8+8k))),{k,0,10}]Product[(1-(q^(16k+6)))(1-(q^(16k+10))),{k,0,10}],{q,0,10}]

Product[(1-(q^(8k+1)))(1-(q^(8k+7)))(1-(q^(8+8k))),{k,0,∞}]Product[(1-(q^(16k+6)))(1-(q^(16k+10))),{k,0,∞}]/QPochhammer[q]