Dixon's Identity/Gaussian Binomial Form/Formulation 1

Theorem

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

Proof

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Source of Name

This entry was named for Alfred Cardew Dixon.

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 $62$