LOG#163. Q-stuff and wonderful functions.

In this blog post I am going to define and talk about some interesting objects. They are commonly referred as q-objects in general.

The q-Pochhammer symbol is the next product:

(1) $\begin{equation*} (a;q)_n=\prod_{k=0}^{n-1}(1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}) \end{equation*}$

with $(a;q)_0\equiv 1$ . The infinite product extension is also very popular

(2) $\begin{equation*} \boxed{(a;q)_\infty=\prod_{k=0}^\infty(1-aq^k)} \end{equation*}$

and it is analytic in the unit disc, with $\phi(q)=(q;q)_\infty$ being the Euler’s function, important object in combinatorics, number theory and the theory of modular forms.

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

The q-Pochhammer symbol satisfies a big number of identities. I like mostly four of them:

(3) $\begin{equation*} \boxed{(a;q)_n=\dfrac{(a;q)_\infty}{(aq^n;q)_\infty}} \end{equation*}$

(4) $\begin{equation*} \boxed{(a;q)_{-n}=\dfrac{1}{(aq^{-n};q)_n}=\prod_{k=1}^n\dfrac{q}{1-aq^{-k}}} \end{equation*}$

(5) $\begin{equation*} \boxed{(a;q)_{-n}=\dfrac{(-q/a)^nq^{n(n-1)/2}}{(q/a;q)_n}} \end{equation*}$

And the fourth is the so-called q-binomial theorem

(6) $\begin{equation*} \boxed{\dfrac{(ax;q)_\infty}{(x;q)_\infty}=\sum_{n=-\infty}^\infty\dfrac{(a;q)_n}{(q;q)_n}x^n} \end{equation*}$

Interpretation: the coefficient of $q^ma^n$ in the expansion

$(a;q)^{-1}=\prod_{k=0}^\infty(1-aq^k)^{-1}=\sum_{k=0}^\infty\dfrac{a^k}{(q;q)_k}$

is the number of partitions of m into at most n parts. Moreover, if we write

$(-a;q)_\infty=\prod_{k=0}^\infty(1+aq^k)$

that is the number of partitions of m into n or n-1 parts when we read off the coefficient in $q^ma^n$ . The q-Pochhammer function admits multiple arguments in the following way:

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

The q-Pochhammer symbol can be related to other q-objects. We define first the q-numbers and the q-factorial. The q-numbers are defined as

$\left[n\right]_q=\dfrac{1-q^n}{1-q}$

and the q-factorial is

$\left[n_q\right]!=\prod_{k=1}^n\left[k\right]_q=\left[1\right]_q\left[2\right]_q\cdots\left[n-1\right]_q\left[n\right]_q=\dfrac{(q;q)_n}{(1-q)^n}$

Now, we can even define a q-deformed version of the traditional derivative. It is called the q-derivative or Jackson’s derivative:

(7) $\begin{equation*} \left(\dfrac{d}{dx}\right)_qf(x)=D_qf(x)=\dfrac{f(qx)-f(x)}{qx-x} \end{equation*}$

It satifies some conventional rules and some deformed variants of the classical derivative

$D_q(f(x)+g(x))=D_qf(x)+D_qg(x)$

$D_q(f\cdot g)(x)=g(x)D_qf(x)+f(qx)D_qg(x)=g(qx)D_qf(x)+f(x)D_qg(x)$

Moreover, we also have

$D_q^nf(0)=\dfrac{f^{(n)}(0)}{n!}\dfrac{(q;q)_n}{(1-q)^n}=\dfrac{f^{(n)}(0)}{n!}\left[n\right]_q$

The Taylor expansion analogue also exists:

$f(z)=\sum_{n=0}f^{(n)}(0)\dfrac{z^n}{n!}=\sum_{n=0}^\infty (D_q^nf)(0)\dfrac{z^n}{\left[n\right]_q!}$

There are also some theta functions to explore. The q-theta function is

(8) $\begin{equation*} \theta(z;q)=\prod_{n=0}^\infty(1-q^nz)\left(1-\frac{q^{n+1}}{z}\right) \end{equation*}$

and with $0\leq \vert q\vert <1$ this yields

(9) $\begin{equation*} \theta(z;q)=\theta\left(\dfrac{q}{z};q\right)=-z\theta\left(\dfrac{1}{z};q\right)=\dfrac{(z;q)_\infty}{\left(\dfrac{q}{z};q\right)_\infty} \end{equation*}$

The Ramanujan theta function is a fascinating object I wish to show you:

(10) $\begin{equation*} \boxed{f(a,b)=\sum_{n=-\infty}^\infty a^{n(n+1)/2}b^{n(n-1)/2}} \end{equation*}$

with $\vert ab\vert <1$ . It appears (and can be used in some applications of) in critical bosonic string theory, superstring theory and M-theory. This Ramanujan theta function satisfies a beautiful identity called Jacobi triple product

$\boxed{f(a,b)=(-a;ab)_\infty(-b;ab)_\infty(ab;ab)_\infty}$

Some additional identities of this Ramanujan theta function are:

(11) $\begin{equation*} f(q;q)=\sum_{n=-\infty}^\infty q^{n^2}=(-q;q^2)_\infty^2(q^2;q^2)_\infty \end{equation*}$

(12) $\begin{equation*} f(q;q^3)=\sum_{n=-\infty}^\infty q^{n(n+1)/2}=(q^2;q^2)_\infty(-q;q)_\infty \end{equation*}$

and

(13) $\begin{equation*} f(-q;-q^2)=\sum_{n=-\infty}^\infty(-1)^nq^{n(3n-1)/2}=(q;q)_\infty=\phi(q) \end{equation*}$

so the Euler $\phi$ function is a particular case of the Ramanujan theta function! They are also related to the Dedekind eta function. The Jacobi theta function $\vartheta$ may be written in terms of the Ramanujan theta function as well:

$\boxed{\vartheta(w;q)=f(wq^2;qw^{-2})}$

By the other hand, the Jacobi triple product identity is generally something more “general”. It is the mathematical identity:

$\prod_{m=1}^\infty\left( 1 - x^{2m}\right)\left( 1 + x^{2m-1} y^2\right)\left(1 +\frac{x^{2m-1}}{y^2}\right)= \sum_{n=-\infty}^\infty x^{n^2} y^{2n}$

and it is defined for arbitrary complex numbers $x,y$ such as $\vert x\vert<1$ and $y\neq 0$ . The two more elegant forms of this Jacobi triple product identity are bound to the Ramanujan theta function we have defined above or in terms of the q-Pochhammer symbols

$\sum_{n=-\infty}^\infty q^{\frac{n(n+1)}{2}}z^n=(q;q)_\infty\left(-\frac{1}{z};q\right)_\infty (-zq;q)_\infty$

where $(a;q)_\infty$ is the infinite q-Pochhammer symbol and

$\sum_{n=-\infty}^\infty a^{\frac{n(n+1)}{2}}b^{\frac{n(n-1)}{2}}= (-a; ab)_\infty (-b; ab)_\infty (ab;ab)_\infty$

in terms of Ramanujan theta function, as we have already seen previously. Aren’t you amazed by those formulae? You should! They are strikingly appealing and beautiful.

May the q-functions be with you!!!!

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The Spectrum Of Riemannium

Mobilis in mobili: Physmatics Chronicles!

LOG#163. Q-stuff and wonderful functions.

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