q-Abel's Theorem

Author

Eric W. Weisstein
December 7, 2008

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

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

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

(∑_(y=0)^m)(((-1))^(m-y)) (1-w (q^m)) (q^(Binomial[m-y, 2])) (QBinomial[m, y, q]) (((((1-w (q^y)))^m) )/(q-w (q^y)))(QPochhammer[-((1-z)/(1-w (q^y))), q, y])

This is from Bhatnagar, p. 105, 1995.

Sum[(((-1))^(m-y))(q^(Binomial[m-y,2]))QBinomial[m,y,q]((1-w (q^m))/(q-w (q^y)))(((1-w (q^y)))^m)QPochhammer[-((1-z)/(1-w (q^y))),q,y],{y,0,m}]

Table[Sum[(((-1))^(m-y))(q^(Binomial[m-y,2]))QBinomial[m,y,q]((1-w (q^m))/(q-w (q^y)))(((1-w (q^y)))^m)QPochhammer[-((1-z)/(1-w (q^y))),q,y],{y,0,m}],{m,1}]//Simplify

Table[Sum[(((-1))^(m-y))(q^(Binomial[m-y,2]))QBinomial[m,y,q]((1-w (q^m))/(q-w (q^y)))(((1-w (q^y)))^m)QPochhammer[-((1-z)/(1-w (q^y))),q,y],{y,0,m}],{m,3}]/.w->0//FullSimplify

Table[(((1-z))^m)(q^(Binomial[m,2])),{m,3}]

Something is wrong here...

FullSimplify[((QPochhammer[q,q,n])/(QPochhammer[q,q,y]QPochhammer[q,q,n-y]))]

FullSimplify[QBinomial[n,y,q]-%]