How can I solve $\sum_{b=0}^{\frac{k}{\sqrt{x}}}\sum_{a=1}^{x} \frac{aF(r\sqrt{a^2+b^2})}{\sqrt{a^2+b^2}}=cF(\delta(x+l))$ for the function $F(x)$, where $k$, $r$, $c$, $\delta$ and $l$ are constants, and $a$, $b$ are indices? Is it possible to solve?
I encountered the problem above while trying to solve a rather difficult physics/mechanics problem, and I am particularly interested in what the graph of $F(x)$ looks like (I am aware that a closed-form solution for $F(x)$ may not exist judging from how ghastly the original equation looks).
I have tried attacking this from many different angles, including an attempt to 'undo' the summations on the left-hand side by applying the property $\sum_{n=a}^{x} f(n) = g(x) \implies f(x) = g(x)-g(x-1)$ twice, but I was left with yet another functional equation that I had no idea how to tackle. I also tried turning the left-hand side into a double integral instead (I guessed that this would perhaps give the approximate shape of the function) and then differentiating both sides twice using the chain rule, but this lead to yet another functional differential equation that I did not know how to solve. In each case, I tried to use a series solution of the form $F(x) = \sum_{n=0}^{\infty} a_n x^n$, my attempt spread across pages of tedious algebra, but to no avail. Needless to say, I'm quite stuck at this stage and don't know how to proceed. Through experimentation on Desmos, I have found that functions of the form $F(x) = x^{-n} - x^{-(n+1)}$ seem to work (where $n$ is a positive integer), but I have no clue as to whether this is correct in general, or how to arrive at such a result.
I would greatly appreciate any tips or insight into the nature of this pretty elusive problem (although an answer may be more obvious to most people than to me).
Thanks