Dixon's Identity/Gaussian Binomial Form

Theorem

Formulation 1

For $l, m, n \in \Z_{\ge 0}$:

$\ds \sum_{k \mathop \in \Z} \paren {-1}^k \dbinom {m - r - s} k_q \dbinom {n + r - s} {n - k}_q \dbinom {r + k} {m + n}_q = \dbinom r m_q \dbinom s n_q$

where $\dbinom r m_q$ denotes a Gaussian binomial coefficient

Formulation 2

For $l, m, n \in \Z_{\ge 0}$:

$\ds \sum_{k \mathop \in \Z} \paren {-1}^k \dbinom {l + m} {l + k}_q \dbinom {m + n} {m + k}_q \dbinom {n + l} {n + k}_q = \dfrac {\paren {l + m + n}!_q} {l!_q \, m!_q \, n!_q}$

where:

$\dbinom {l + m} {l + k}_q$ denotes a Gaussian binomial coefficient

$l!_q$ is defined as $\ds \prod_{k \mathop = 1}^n \paren {1 + q + \cdots + q^{k - 1} }$

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 $62$