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q-Binomial Theorem

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The q-analog of the binomial theorem

(1-z)^n=1-nz+(n(n-1))/(1·2)z^2-(n(n-1)(n-2))/(1·2·3)z^3+...
(1)

is given by

(1-z/(q^n))(1-z/(q^(n-1)))...(1-z/q) =1-(1-q^n)/(1-q)z/(q^n)+(1-q^n)/(1-q)(1-q^(n-1))/(1-q^2)(z^2)/(q^(n+(n-1))) -...+/-(z^n)/(q^(n(n+1)/2)).
(2)

Written as a q-series, the identity becomes

sum_(n=0)^(infty)((a;q)_n)/((q;q)_n)z^n=((az;q)_infty)/((z;q)_infty)
(3)
=_1phi_0(a;;q,z),
(4)

where (a;q)_n is a q-Pochhammer symbol and _rphi_s(a_1,...;b_2,...;q,z) is a q-hypergeometric function (Heine 1847, p. 303; Andrews 1986). The Cauchy binomial theorem is a special case of this general theorem.

See also

Binomial Series, Binomial Theorem, Cauchy Binomial Theorem, Ramanujan Psi Sum

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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 1+((1-q^alpha)(1-q^beta))/((1-q)(1-q^gamma))·x+((1-q^alpha)(1-q^(alpha+1))(1-q^beta)(1-q^(beta+1)))/((1-q)(1-q^2)(1-q^gamma)(1-q^(gamma+1)))·x^2+...." 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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q-Binomial Theorem

Cite this as:

Weisstein, Eric W. "q-Binomial Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/q-BinomialTheorem.html

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