Q-series

(a;q)_n = \prod_{k=0}^{n-1} (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_{k=1}^\infty (1-q^k)

is known as Euler's Function , and is important in Combinatorics , Number Theory , and the theory of Modular Forms .

IDENTITIES

Letting

(a;q)_\infty

stand for the infinite product, one has

:

(a;q)_n = rac{(a;q)_\infty} {(aq^n;q)_\infty}

which extends the definition to negative integers ''n''. Thus, for

n\ge 0

, one has

:

(a;q)_{-n} = rac{1}{(aq^{-n};q)_n}

and

:

(a;q)_{-n} = rac{(-q/a)^n q^{n(n-1)/2}} {(q/a;q)_n}

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_{q
ightarrow 1^-}rac{1-q^n}{1-q}=n,

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

:

{Link without Title} _q=rac{1-q^n}{1-q}.

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

	<math>	1(1+q)\cdots (1+q+\cdots + q^{n-2}) (1+q+\cdots + q^{n-1})</math>
	<math>	rac{(qq)_n}{(1-q)^n}</math>

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