TITLE: A q-analogue of Ramanujan's tau function
QUESTION [8 upvotes]: There have been a couple of questions on Ramanujan's $\tau$ function.
Lehmer's conjecture for Ramanujan's tau function
The Vanishing of Ramanujan's Function tau(n)
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?

REPLY [4 votes]: 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).