Question

There have been a couple of questions on Ramanujan's $\tau$ function.
http://mathoverflow.net/questions/32620/
http://mathoverflow.net/questions/31058/

A $q$-analogue is given by $$ \tau_q(n) = \sum_{|\lambda|=n}\prod_{(i,j)} \frac{[5-h(i,j)][5+h(i,j)]}{[h(i,j)]^2} $$

Here $[k]=\frac{q^k-q^{-k}}{q-q^{-1}}$. The sum is over all partitions of $n$. The product is over all boxes in the diagram of $\lambda$ and $h(i,j)$ is the hook length of the box $(i,j)$.

This is a $q$-analogue because (despite appearances) it is a Laurent polynomial in $q$ and substituting $q=1$ gives $\tau(n)$.

It would be ridiculous to ask if this is $q$-holonomic.

Is this $q$-analogue known? and does it have any significance in number theory?

Answer 1

I am not very well familiar with the notion "$q$-holonomic" but learned it quite recently from this question. If the sequence $\tau_q(n)$ were $q$-holonomic, then $\tau(n)$ would be holonomic. The latter means that $$\Delta(x)=x\prod_{n=1}^\infty(1-x^n)^{24}=\sum_{k=1}^\infty\tau(k)x^k$$ satisfies a linear differential equation with rational function coefficients. However, this is known to be false: the function $\Delta(x)$ has singularities at all roots of unity.

Note that the function $\Delta(x)$ satisfies an algebraic differential equation with constant coefficients; the result due to Halphen (and independently, although later, by Ramanujan).