I'm trying to solve a fifth order recurrence relation with non-constant coefficients:
$$ \begin{multline}[\xi_0(j+1)+\pi_1j(j+1)]c_{j+1} + [\rho_0+\xi_1j+\pi_2j(j-1)]c_j\\+[\rho_1+\xi_2(j-1)+\pi_3(j-1)(j-2)]c_{j-1}+[\rho_2+\xi_3(j-2)+\pi_4(j-2)(j-3)]c_{j-2}\\+[\rho_3+\xi_4(j-3)+\pi_5(j-3)(j-4)]c_{j-3}+[\xi_5(j-4)+\pi_6(j-4)(j-5)]c_{j-4}=0 \end{multline}$$
where the $\pi_i$, $\xi_i$, and $\rho_i$, are known (non-integer) constants, and I also know any necessary initial conditions $c_i$ in terms of $c_0$ and the other constants. This relation arose from a power series solution to a non-linear second-order differential equation with $6$ regular singularities.
I've tried using the substitution $b_{j+1}=(j+1)c_{j+1}$ but the $\rho_i$ prevent this from working. The form certainly looks complicated, but since each term depends only on its index and the previous index, I'm hopeful of a solution.
Does anyone know any tricks which might be useful to re-write this expression with constant coefficients? Clearly factorials will play a role here.
Some sources I've already looked at:
https://www.csee.umbc.edu/~stephens/203/PDF/8-3.pdf https://www.math.ust.hk/~mabfchen/Math2343/Recurrence.pdf https://math.la.asu.edu/~dajones/class/275/ch7.pdf
EDIT: While I have not been able to solve this relation exactly, I am able to study asymptotic behavior using the Poincare-Perron theorem. This gives an asymptotic constant-coefficient characteristic equation
$$\pi_1\lambda^5+\pi_2\lambda^4+\pi_3\lambda^3+\pi_4\lambda^2+\pi_5\lambda+\pi_6=0$$
which, assuming the roots of this polynomial have distinct moduli $|\lambda_i|$, indicates the power series associated with this equation will converge with a radius $|x|<|\lambda_{max}|^{-1}$.