LOG#163. Q-stuff and wonderful functions.

RamanujanSeal

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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