Gaussian Binomial Theorem/Real Numbers

Theorem

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

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

where:

$\dbinom r k_q$ denotes a Gaussian binomial coefficient

$x \in \R: \size x < 1$

$q \in \R: \size q < 1$.

Proof

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Source of Name

This entry was named for Carl Friedrich Gauss.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $58$