PlanetMath: Gaussian polynomials

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Gaussian polynomials

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For an indeterminate and integers $n \ge m \ge 0$ we define the following:

(a) $(m)_q = q^{m-1} + q^{m-2} + \cdots + 1$ for ,

(b) $(m!)_q = (m)_q (m-1)_q \cdots (1)_q$ for , and ,

The expressions ${n \choose m}_q$ are called -binomial coefficients or Gaussian polynomials.

Note: if we replace with 1, then we obtain the familiar integers, factorials, and binomial coefficients. Specifically,

(a) ,

(b) ,

(d) ${m \choose m}_q=1$ .

"Gaussian polynomials" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Other names:

q-binomial coefficients

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Cross-references: binomial coefficients, factorials, coefficients, expressions, integers, indeterminate
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This is version 5 of Gaussian polynomials, born on 2001-10-19, modified 2006-02-18.
Object id is 378, canonical name is GaussianPolynomials.
Accessed 3756 times total.

Classification:

AMS MSC:	16S36 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Ordinary and skew polynomial rings and semigroup rings)
	05A10 (Combinatorics :: Enumerative combinatorics :: Factorials, binomial coefficients, combinatorial functions)
	05A30 (Combinatorics :: Enumerative combinatorics :: $q$-calculus and related topics)

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