Gaussian binomial

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In mathematics, the Gaussian binomials (sometimes called the Gaussian coefficients, or the q-binomial coefficients) are the q-analogs of the binomial coefficients.

The Gaussian binomials are defined by

<math>{m \choose r}_q

= \frac{(1-q^m)(1-q^{m-1})\cdots(1-q^{m-r+1})} {(1-q)(1-q^2)\cdots(1-q^r)}.</math>

One can prove that

<math>\lim_{q\rightarrow 1^-} {m \choose r}_q =

{m \choose r}.</math>

The Pascal identities for the Gaussian binomials are

<math>{m \choose r}_q = q^r {m-1 \choose r}_q + {m-1 \choose r-1}_q</math>

and

<math>{m \choose r}_q = {m-1 \choose r}_q + q^{m-r}{m-1 \choose r-1}_q.</math>

The Newton binomial formulas are

<math>\prod_{k=0}^{n-1} (1+q^kt)=\sum_{k=0}^n q^{k(k-1)/2}

{n \choose k}_q t^k </math>

and

<math>\prod_{k=0}^{n-1} \frac{1}{(1-q^kt)}=\sum_{k=0}^\infty

{n+k-1 \choose k}_q t^k. </math>

Like the ordinary binomial coefficients, the Gaussian binomials are center-symmetric i.e. invariant under the reflection <math> r \rightarrow m-r </math>:

<math>{m \choose r}_q = {m \choose m-r}_q. </math>

The first Pascal identity allows one to compute the Gaussian binomials recursively (with respect to m ) using the initial "boundary" values

<math>{m \choose m}_q ={m \choose 0}_q=1 </math>

and also incidentally shows that the Gaussian binomials are indeed polynomials (in q). The second Pascal identity follows from the first using the substitution <math> r \rightarrow m-r </math> and the invariance of the Gaussian binomials under the reflection <math> r \rightarrow m-r </math>. Both Pascal identities together imply

<math>{m \choose r}_q = {{1-q^{m}}\over {1-q^{m-r}}} {m-1 \choose r}_q </math>

which leads (when applied iteratively for m, m − 1, m − 2,....) to an expression for the Gaussian binomial as given in the definition above.

Applications

Gaussian binomials occur in the counting of symmetric polynomials and in the theory of partitions. The also play an important role in the enumerative theory of symmetric spaces defined over a finite field. In particular, for every finite field F_{q with q elements, the Gaussian binomial}

<math>{n \choose k}_q</math>

counts the number v_{n,k;q of different k-dimensional vector subspaces of an n-dimensional vector space over Fq (a Grassmannian). For example, the Gaussian binomial}

<math>{n \choose 1}_q=1+q+q^2+\cdots+q^{n-1}</math>

is the number of different lines in F_{q^{n (a projective space).}}

References

• Eugene Mukhin, Symmetric Polynomials and Partitions (undated, 2004 or earlier).
• Ratnadha Kolhatkar, Zeta function of Grassmann Varietes (dated January 26, 2004)