Q-series

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In mathematics, in the area of combinatorics, a q-series, also sometimes called a q-shifted factorial, is a q-analog of the common factorial. It is defined as

[(a;q)_n = \prod_^ (1-aq^k).]

It is usually considered first as a formal power series; it is also an analytic function of q, in the unit disc. The function

[\phi(q) = (q;q)_\infty=\prod_^\infty (1-q^k)]

is known as Euler's function, and is important in combinatorics, number theory, and the theory of modular forms.

Contents

1 Identities
2 Multiple variables
3 Relationship to the ''q''-bracket and the ''q''-binomial
4 See also
5 References

Identities

Letting [(a;q)_\infty] stand for the infinite product, one has

[(a;q)_n = \frac ]

which extends the definition to negative integers n. Thus, for [n\ge 0], one has

[(a;q)_ = \frac;q)_n}]

and

[(a;q)_ = \frac} ]

Multiple variables

The notation

[(a_1,a_2,\ldots,a_m;q)_n = (a_1;q)_n (a_2;q)_n \ldots (a_m;q)_n]

is often used to denote a q-series for multiple variables.

Relationship to the q-bracket and the q-binomial

For convenience, the limit q → 1 inside the unit circle is written as the limit q → 1⁻, which suggests the limit through real values tending up to 1, which is in fact more restricted, though the difference is not usually significant.

Noticing that

[\lim_\frac=n,]

we define the q-analog of n, also known as the q-bracket of n to be

[[n]_q=\frac.]

From this one can define the q-analog of the factorial, the q-factorial, as

[\big[n]_q!]	[=[1]_q [2]_q \cdots [n-1]_q [n]_q]
	[=\frac \frac \cdots \frac} \frac]
	[=1(1+q)\cdots (1+q+\cdots + q^) (1+q+\cdots + q^)]
	[=\frac.]

Again, one recovers the usual factorial by taking the limit as [q\rightarrow 1^].

From the q-factorials, one can move on to define the q-binomial coefficients, also known as Gaussian coefficients, Gaussian polynomials, or Gaussian binomials:

[\beginn\\k\end_q=\frac.]

References

Eugene Mukhin, [Symmetric Polynomials and Partitions] (undated, 2004 or earlier)
[q-analog] from MathWorld
[q-bracket] from MathWorld
[q-factorial] from MathWorld
[q-binomial coefficient] from MathWorld

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Q-series

Identities

Multiple variables

Relationship to the q-bracket and the q-binomial

See also

References