I am concerned in finding an explicit solution to the following functional equation, where $P$ is the unknown function:
$$P(q) = \lambda_1(q)+ \lambda_2(q)P(q+2 \chi_s),$$
where: $$\lambda_1(q)=\frac{\alpha_2(q) + \alpha_2(q+2\chi_s)}{\cosh q - \alpha_1(q)}$$ and: $$\lambda_2(q) = \frac{\cosh(q+2\chi_s)+\alpha_1(q+2\chi_s)}{\cosh q - \alpha_1(q)}$$ he auxiliary functions $\alpha_1$ and $\alpha_2$, are explicitly given by:
$$\alpha_1(q)= \alpha_{10}\sinh q + \frac{\alpha_{11}}{\sinh q},\ \ \ \alpha_2(q)= \frac{\alpha_{20}}{\sinh q}.$$
Besides, the real numbers $\alpha_{10}$, $\alpha_{11}$, $\alpha_{20}$ and $\chi_s$ are initial parameters of the problem, and are given real constants.
Iterating the original functional equation gives us a particular solution for P(q), expressed as an infinite series of the form:
$$P(q)= \lambda_1(q)+ \sum_{n=1}^{\infty}\lambda_1(q+2n\chi_s)\prod_{j=0}^{n-1}\lambda_2(q+2j\chi_s)$$
which can be verified to be a solution after substituting in the original equation. However, I would like to obtain an explicit solution written with a finite number of terms, not an infinite series. I wonder if this is actually possible, given its complexity. Any hint will be certainly acknowledged.
A direct approch to solving the functional equation is by defining an iteration sequence, like : ($n\geq 1$, is an integer number defining the step level inside the iteration sequence) \begin{equation} P^{[n]}(q)= \lambda_1(q)+ \lambda_2(q)P^{[n-1]}(q+ 2\chi_s) \end{equation}
which is quite good for numerical evaluation, once we decide the seed function $P^{[0]}(q)$ for which, we usually choose:
$$P^{[0]}(q)=\frac{\lambda_1(q)}{1-\lambda_2(q)},$$
Unfortunately, the sequence has to be truncated at some high value n of the iteration index and the solution so obtained is still not very useful, because for certain initial conditions of the problem (which would take me way too far in the description of the physical phenomenon), giving rise to the above functional equation, very high values of n might be needed into obtain an accurate enough representation of P(q). This the reason why I struggle trying to get a finite form solution to the equation, expressible with a finite number of operations on the functions $\lambda_{1,2}(q)$ [1].
Hope this is doable.
Thanks again for your attention.
[1] "Analytical asymptotic velocities in linear Richtmyer-Meshkov-like flows" by F. Cobos Campos and J. G. Wouchuk, Physical Review E 90, 053007 (2014).