Calculating the density of solutions of equations related to the P\'olya-Ostrowski group through Markov chains

Motivated by a problem in the theory of integer-valued polynomials, we investigate the natural density of the solutions of equations of the form $\theta_uu_q(n)+\theta_ww_q(n)+\theta_2\frac{n(n+1)}{2}+\theta_1n+\theta_0\equiv 0\bmod d$, where $d,q\geq 2$ are fixed integers, $\theta_u,\theta_w,\theta_2,\theta_1,\theta_0$ are parameters and $u_q$ and $w_q$ are functions related to the $q$-adic valuations of the numbers between 1 and $n$. We show that the number of solutions of this equation in $[0,N)$ satisfies a recurrence relation, with which we can associate to any pair $(d,q)$ a stochastic matrix and a Markov chain. Using this interpretation, we calculate the density for the case $\theta_u=\theta_2=0$ and for the case $\theta_u=1$, $\theta_w=\theta_2=\theta_1=0$ and either $d|q$ or $d$ and $q$ are coprime.