Addition Rule for Gaussian Binomial Coefficients

Theorem

Let $\dbinom n m_q$ denote a Gaussian binomial coefficient.

Formulation 1

$\dbinom n m_q = \dbinom {n - 1} m_q + \dbinom {n - 1} {m - 1}_q q^{n - m}$

Formulation 2

$\dbinom n m_q = \dbinom {n - 1} m_q q^m + \dbinom {n - 1} {m - 1}_q$

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 $58$