Q-binomial theorem

In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients. The Gaussian binomial coefficient $\textstyle {\binom {n}{k}}_{q}$ $\textstyle {\binom {n}{k}}_{q}$ is a polynomial in q with integer coefficients, whose value when q is set to a prime power counts the number of subspaces of dimension k in a vector space of dimension n over a finite field with q elements.

The Gaussian binomial coefficients are defined by

where m and r are non-negative integers. For r = 0 the value is 1 since numerator and denominator are both empty products. Although the formula in the first clause appears to involve a rational function, it actually designates a polynomial, because the division is exact in Z[q]. Note that the formula can be applied for r = m + 1, and gives 0 due to a factor 1 − q⁰ = 0 in the numerator, in accordance with the second clause (for even larger r the factor 0 remains present in the numerator, but its further factors would involve negative powers of q, whence explicitly stating the second clause is preferable). All of the factors in numerator and denominator are divisible by 1 − q, with as quotient a q number:

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