q-Binomial Theorem

The q-Analog of the Binomial Theorem

$$(1-z)^n=1-nz+{n(n-1)\over 1\cdot 2}z^2-{n(n-1)(n-2)\over 1\cdot 2\cdot 3}z^3+\ldots$$

is given by

$\left({1-{z\over q^n}}\right)\left({1-{z\over q^{n-1}}}\right)\cdots\left({1-{z\over q}}\right)$
$= 1-{1-q^n\over 1-q}{z\over q^n}+{1-q^n\over 1-q}{1-q^{n-1}\over 1-q^2}{z^2\over q^{n+(n-1)}}-\ldots\pm{z^n\over q^{n(n+1)/2}}.$

Written as a q-Series, the identity becomes

$$\sum_{n=0}^\infty {(a;q)_n\over (q;q)_n} z^n={(az;q)_\infty\over (z;q)_\infty},$$

where

$$(a;q)_n=\prod_{m=0}^\infty {(1-aq^m)\over(1-aq^{m+n})}$$

(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, Heine Hypergeometric Series, Ramanujan Psi Sum

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.

Heine, E. ``Untersuchungen über die Reihe $1+{(1-q^\alpha)(1-q^\beta)\over(1-q)(1-q^\gamma)}\cdot x+{(1-q^\alpha)(1-q^{\al... ...(1-q^{\beta+1})\over (1-q)(1-q^2)(1-q^\gamma)(1-q^{\gamma+1})}\cdot x^2+\ldots$.'' J. reine angew. Math. 34, 285-328, 1847.