Gaussian Binomial Theorem/Negation of Upper Index

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Theorem

Let $r \in \R$ be a real number.


\(\ds \prod_{k \mathop \ge 0} \dfrac {1 + q^{k + r + 1} x} {1 + q^k x}\) \(=\) \(\ds \sum_{k \mathop \in \Z} \dbinom {-r - 1} k_q q^{k \left({k - 1}\right) / 2} \left({-q^{r + 1} x}\right)^k\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop \in \Z} \dbinom {k + r} x^k\)

where:

$\dbinom r k_q$ denotes a Gaussian binomial coefficient.
$x \in \R: \left\lvert{x}\right\rvert < 1$
$q \in \R: \left\lvert{q}\right\rvert < 1$.


Proof




Source of Name

This entry was named for Carl Friedrich Gauss.


Sources