Chemistry - Q-series

In mathematics, a q-series is defined as

$(a,q)_n = \prod_{k=0}^{n-1} (1-aq^k)$

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

The Euler function is given by

$\phi(q)=\prod_{k=1}^\infty (1-q^k)$

The coefficient of $q k$ in the Maclaurin series for $1 / φ(q)$ gives the number of all partitions of k. That is,

$\frac{1}{\phi(q)}=\sum_{k=0}^\infty p(k) q^k$

where $p (k)$ is the partition function of k.

The Euler identity is

$\phi(q)=\sum_{n=-\infty}^\infty (-)^n q^{(3n^2-n)/2}$

Note that $(3 n 2 - n) / 2$ is a pentagon number.

The Euler function is related to the Dedekind eta function through a Ramanujan identity as

φ(q) = q - 1 / 24 η(τ)

where $q = e 2π i τ$ is the square of the nome.

Note that both functions have the symmetry of the modular group. The Euler function also plays a role in describing the interior of the Mandelbrot set.

Contents

Q-analogs

There is a substantial theory constructing q-analogs of results, in particular in combinatorics and the theory of special functions. A q-analog, roughly speaking, is a theorem or identity for a q-series that gives back a known result in the limit, as q → 1, inside the unit circle. For convenience this 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

$\lim_{q\rightarrow 1^-}\frac{1-q^n}{1-q}=n,$

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

$[n]_q=\frac{1-q^n}{1-q}.$

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

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

Again, one recovers the usual factorial by taking the limit as $q\rightarrow 1^{-1}$ .

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

$\begin{bmatrix} n\\ k \end{bmatrix}_q = \frac{[n]_q!}{[n-k]_q! [k]_q!}.$

References

Eugene Mukhin Symmetric Polynomials and Partitions (undated, 2004 or earlier)
Linas Vepstas Spectral Analysis of Mandelbrot Interior (2000) describes the appearance of q-series in the Mandelbrot Set.