In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients.
Definition
The Gaussian binomial coefficients are defined by
:{m \choose r}_q
= \begin{cases}
\frac{(1-q^m)(1-q^{m-1})\cdots(1-q^{m-r+1})} {(1-q)(1-q^2)\cdots(1-q^r)} & r \le m \\
0 & r>m \end{cases}
where m and r are non-negative integers. For 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 d...
Roughly speaking, in mathematics, specifically in the areas of combinatorics and special functions, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1. Typically, mathematicians are interested in q-analogues that arise naturally, rather than in arbitrarily contriving q-analogues of known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.
q-analogs find applications in a number of area...
