q-Gauss Identity

Author

Eric W. Weisstein
April 4, 2011

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

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

©2011 Wolfram Research, Inc. except for portions noted otherwise

(∑_(k=0)^n)(q^(k^2)) ((((QBinomial[n, k, q])))^2)==(((QPochhammer[(Sqrt[q]), q, n]) (QPochhammer[-(Sqrt[q]), q, n]) (QPochhammer[-q, q, n]))/(QPochhammer[q, q, n]))

V7

TimeConstrained[Sum[(q^(k^2))((QBinomial[n,k,q])^2),{k,0,n}],120]//Timing

Subtract@@@Table[{NSum[(q^(k^2))((QBinomial[n,k,q])^2),{k,0,n}],(QPochhammer[(Sqrt[q]),q,n]QPochhammer[-(Sqrt[q]),q,n]QPochhammer[-q,q,n])/QPochhammer[q,q,n]}/.{q->RandomReal[],n->RandomInteger[{1,10}]},{100}]//Chop//Union//Quiet

V8.1

$Version

TimeConstrained[Sum[(q^(k^2))((QBinomial[n,k,q])^2),{k,0,n}],120]//Timing

Subtract@@@Table[{NSum[(q^(k^2))((QBinomial[n,k,q])^2),{k,0,n}],((QPochhammer[-q,q,n] QPochhammer[q,(q^2),n])/(QPochhammer[q,q,n]))}/.{q->RandomReal[],n->RandomInteger[{1,10}]},{100}]//Chop//Union//Quiet