TOPICS
The 
-binomial
theorem is the q-analog of the binomial
theorem
 |
(1)
|
is given by
 |
(2)
|
Written as a q-series, the identity becomes
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is a 
-Pochhammer
symbol and 
is a 
-hypergeometric
function (Heine 1847, p. 303; Andrews 1986). The Cauchy
binomial theorem is a special case of this general theorem.
Extensions." PhD thesis. Columbus, OH: Ohio State
University, p. 24, 1995.Gasper, G. "Elementary Derivations
of Summation and Transformation Formulas for q-Series." In Fields
Inst. Comm. 14 (Ed. M. E. H. Ismail et al. ), pp. 55-70,
1997.Gasper, G. and Rahman, M. Basic
Hypergeometric Series. Cambridge, England: Cambridge University Press, p. 7,
1990.Heine, E. "Untersuchungen über die Reihe 
."
J. reine angew. Math. 34, 285-328, 1847.Koepf, W. Hypergeometric
Summation: An Algorithmic Approach to Summation and Special Function Identities.
Braunschweig, Germany: Vieweg, p. 26, 1998.Weisstein, Eric W. "q-Binomial Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/q-BinomialTheorem.html
