Let $SRMI_q(2n)$ denote the number of self-reciprocal irreducible monic polynomials of even degree $2n$ over the finite field $\mathbf{F}_q$ with $q$ elements. According to Thm 3.1.20 of Handbook of Finite Fields by Mullen and Panario we have that \begin{equation*} SRMI_q(2n) = \frac{1}{2n}\sum_{\text{odd $d \mid n$}} \mu(d)(q^{n/d}-1), \qquad SRMI_q(2n) = \frac{1}{2n}\sum_{\text{odd $d \mid n$}} \mu(d)q^{n/d} \end{equation*} depending on whether $q$ is odd or even. The sums range over all odd divisors of $n$. Now comes my question. Is it possible to write the infinite product \begin{equation*} \prod_{n \geq 1} \frac{1}{(1-x^{2n})^{SRMI_q(2n)}} \end{equation*} as a rational function?

What I have in mind is an analogue of the well-known formula \begin{equation*} \prod_{n \geq 1} \frac{1}{(1-x^n)^{I_q(n)}} = \frac{1}{1-qx} \end{equation*} where $I_q(n)$ is the number of irreducible monic polynomials over $\mathbf{F}_q$.

Let $SRMI_q(2n)$ denote the number of self-reciprocal irreducible monic polynomials of even degree $2n$ over the finite field $\mathbf{F}_q$ with $q$ elements. Recall that a polynomial $p(x) \in \mathbf{F}_q[x]$ of degree $n$ is self-reciprocal if $p(x)=x^np(x^{-1})$. According to Thm 3.1.20 of Handbook of Finite Fields by Mullen and Panario we have that \begin{equation*} SRMI_q(2n) = \frac{1}{2n}\sum_{\text{odd $d \mid n$}} \mu(d)(q^{n/d}-1), \qquad SRMI_q(2n) = \frac{1}{2n}\sum_{\text{odd $d \mid n$}} \mu(d)q^{n/d} \end{equation*} depending on whether $q$ is odd or even. The sums range over all odd divisors of $n$. Now comes my question. Is it possible to write the infinite product \begin{equation*} \prod_{n \geq 1} \frac{1}{(1-x^{2n})^{SRMI_q(2n)}} \end{equation*} as a rational function?

What I have in mind is an analogue of the well-known formula \begin{equation*} \prod_{n \geq 1} \frac{1}{(1-x^n)^{I_q(n)}} = \frac{1}{1-qx} \end{equation*} where $I_q(n)$ is the number of irreducible monic polynomials over $\mathbf{F}_q$.