Gaussian Binomial Theorem

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Theorem

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

\(\displaystyle \sum_{k \mathop \in \Z} \dbinom n k_q q^{k \paren {k - 1} / 2} x^k\) \(=\) \(\displaystyle \prod_{k \mathop = 1}^n \paren {1 + q^{k - 1} x}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {1 + x} \paren {1 + q x} \paren {1 + q^2 x} \cdots \paren {1 + q^{n - 1} x}\)

where $\dbinom n k_q$ denotes a Gaussian binomial coefficient.


Real Numbers

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

$\displaystyle \sum_{k \mathop \in \Z} \dbinom r k_q q^{k \left({k - 1}\right) / 2} x^k = \prod_{k \mathop \ge 0} \dfrac {1 + q^k x} {1 + q^{r + k} x}$

where:

$\dbinom r k_q$ denotes a Gaussian binomial coefficient.
$x \in \R: \left\lvert{x}\right\rvert < 1$
$q \in \R: \left\lvert{q}\right\rvert < 1$.


Negation of Upper Index

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


\(\displaystyle \prod_{k \mathop \ge 0} \dfrac {1 + q^{k + r + 1} x} {1 + q^k x}\) \(=\) \(\displaystyle \sum_{k \mathop \in \Z} \dbinom {-r - 1} k_q q^{k \left({k - 1}\right) / 2} \left({-q^{r + 1} x}\right)^k\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \in \Z} \dbinom {k + r} x^k\)

where:

$\dbinom r k_q$ denotes a Gaussian binomial coefficient.
$x \in \R: \left\lvert{x}\right\rvert < 1$
$q \in \R: \left\lvert{q}\right\rvert < 1$.


Proof

The proof proceeds by induction.

For all $n \in \Z_{\ge 1}$, let $\map P n$ be the proposition:

$\displaystyle \prod_{k \mathop = 1}^n \paren {1 + q^{k - 1} x} = \sum_{k \mathop \in \Z} \dbinom n k_q q^{k \paren {k - 1} / 2} x^k$


Basis for the Induction

$\map P 1$ is the case:

\(\displaystyle \prod_{k \mathop = 1}^1 \paren {1 + q^{k - 1} x}\) \(=\) \(\displaystyle 1 + x\)
\(\displaystyle \sum_{k \mathop \in \Z} \dbinom 1 k_q q^{k \paren {k - 1} / 2} x^k\) \(=\) \(\displaystyle \dbinom 1 0_q q^{0 \paren {0 - 1} / 2} x^0 + \dbinom 1 1_q q^{1 \paren {1 - 1} / 2} x^1\) Gaussian Binomial Coefficient of 1: for all other $k$, $\dbinom 1 k_q = 0$
\(\displaystyle \) \(=\) \(\displaystyle q^{0 \paren {-1} / 2} + q^{1 \times 0 / 2} x^1\) Gaussian Binomial Coefficient of 1: $\dbinom 1 0_q = \dbinom 1 1_q = 0$
\(\displaystyle \) \(=\) \(\displaystyle q^0 + q^0 x\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + x\)


Thus $\map P 1$ is seen to hold.


This is the basis for the induction.


Induction Hypothesis

Now it needs to be shown that, if $\map P r$ is true, where $r \ge 1$, then it logically follows that $\map P {r + 1}$ is true.


So this is the induction hypothesis:

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


from which it is to be shown that:

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


Induction Step

This is the induction step:

\(\displaystyle \prod_{k \mathop = 1}^{r + 1} \paren {1 + q^{k - 1} x}\) \(=\) \(\displaystyle \prod_{k \mathop = 1}^r \paren {1 + q^{k - 1} x} \times \paren {1 + q^r x}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \in \Z} \dbinom r k_q q^{k \paren {k - 1} / 2} x^k \times \paren {1 + q^r x}\) Induction Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \in \Z} \dbinom r k_q q^{k \paren {k - 1} / 2} x^k + q^r x \sum_{k \mathop \in \Z} \dbinom r k_q q^{k \paren {k - 1} / 2} x^k\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \in \Z} \dbinom r k_q q^{k \paren {k - 1} / 2} x^k + \sum_{k \mathop \in \Z} \dbinom r k_q q^r q^{k \paren {k - 1} / 2} x^{k + 1}\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \in \Z} \dbinom r k_q q^{k \paren {k - 1} / 2} x^k + \sum_{k \mathop \in \Z} \dbinom r {k - 1}_q q^r q^{\paren {k - 1} \paren {k - 2} / 2} x^k\) Translation of Index Variable of Summation
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \in \Z} \paren {\dbinom r k_q q^{k \paren {k - 1} / 2} + q^r q^{-\paren {k - 1} } \dbinom r {k - 1}_q q^{k \paren {k - 1} / 2} } x^k\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \in \Z} \paren {\dbinom r k_q + q^{r + 1 - k} \dbinom r {k - 1}_q} q^{k \paren {k - 1} / 2} x^k\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop \in \Z} \dbinom {r + 1} k_q q^{k \paren {k - 1} / 2} x^k\) Addition Rule for Gaussian Binomial Coefficients

So $\map P r \implies \map P {r + 1}$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall n \in \Z_{\ge 1}: \displaystyle \prod_{k \mathop = 1}^n \paren {1 + q^{k - 1} x} = \sum_{k \mathop \in \Z} \dbinom n k_q q^{k \paren {k - 1} / 2} x^k$

$\blacksquare$


Also known as

Some sources give this as the $q$-nomial theorem.

Some refer to it as the $q$-binomial theorem.


Source of Name

This entry was named for Carl Friedrich Gauss.


Sources