q-Binomial Theorem
المؤلف:
Andrews, G. E.
المصدر:
q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc
الجزء والصفحة:
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26-8-2019
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q-Binomial Theorem
The q-analog of the binomial theorem
 |
(1)
|
is given by
 |
(2)
|
Written as a q-series, the identity becomes
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.
REFERENCES:
Andrews, G. E. q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Providence, RI: Amer. Math. Soc., p. 10, 1986.
Bhatnagar, G. Inverse Relations, Generalized Bibasic Series, and their U(n) Extensions. Ph.D. thesis. 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.
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