All-order resurgent asymptotics for the third-order resonant renormalon

Let $a^{(m)}_{i,j}(n)$ be the coefficients in the trans-series associated to the third-order problem, with $m,i,j\geq 0$ and $s=m-2i-3j\geq 0$. Assume that $a^{(m')}_{i',j'}(n)=0$ if $m'-2i'-3j'<0$ or any of $m',i',j'$ is negative. Let $\psi(z)$ be the derivative of $\log\Gamma(z)$ and define

$$Q(z)=\left(\tfrac{21265}{4608}\right)^2\left(\psi^2(z)+\psi^\prime(z)\right)+2c_1\left(\tfrac{21265}{4608}\right)\psi(z)+c_2.$$

All-order resurgent asymptotics. The coefficients satisfy the asymptotic expansion

$$\begin{aligned} a^{(m)}_{i,j}(n)&\sim -(s+1)S_1\sum_{k\ge0}a^{(m+1)}_{i,j}(k)\Gamma\left(n+\tfrac{35}{12}-k\right)\\ &\quad+S_1\sum_{k\ge0}\left(4(i+1)a^{(m+1)}_{i+1,j}(k)+6(j+1)a^{(m+1)}_{i,j+1}(k)\right)\Gamma\left(n-\tfrac{25}{12}-k\right)\left(\tfrac{21265}{4608}\psi\left(n-\tfrac{25}{12}-k\right)+d_1\right)\\ &\quad+\tfrac14S_3\sum_{k\ge0}\left(4(s+1)a^{(m-1)}_{i-1,j}(k)+6(j+1)a^{(m-1)}_{i-2,j+1}(k)\right)(-1)^{n-k}\Gamma\left(n+\tfrac{25}{12}-k\right)\\ &\quad-2(s-2i-1)S_3\sum_{k\ge0}a^{(m-1)}_{i,j}(k)(-1)^{n-k}\Gamma\left(n-\tfrac{35}{12}-k\right)\left(\tfrac{21265}{4608}\psi\left(n-\tfrac{35}{12}-k\right)+f_1\right)\\ &\quad-S_3\sum_{k\ge0}\left(8(i+1)a^{(m-1)}_{i+1,j}(k)+6(j+1)a^{(m-1)}_{i,j+1}(k)\right)(-1)^{n-k}\Gamma\left(n-\tfrac{95}{12}-k\right)Q\left(n-\tfrac{95}{12}-k\right)\\ &\quad-(f_1-c_1)S_3\sum_{k\ge0}\left(2(i+1)a^{(m-1)}_{i+1,j-1}(k)+6(i+j)a^{(m-1)}_{i,j}(k)\right)(-1)^{n-k}\Gamma\left(n-\tfrac{35}{12}-k\right). \end{aligned}$$

The constants are invariants of the ODE and encode the connection problem between low-order perturbative data and large-order asymptotics. The result is presented as a well-tested solution of the analytic continuation problem at all orders, obtained using empirical and analytical methods.