How to solve a linear algebraic complex equation in one function evaluated at different arguments?

Question

Hello,

I am trying to solve an equation of the form

$C_1 f(k_1 z) + C_2 f(k_2 z) + C_3 f(k_3 z) + C_4 f(k_4 z) = C_5 z^2$

for $f(z)$. Everything is complex. The $C_i$'s and $k_i$'s depend on some other parameters $\{ \lambda_i \}$ through a very complicated dependence.

Any ideas?

Thanks!

Answer 1

Make the change of the independent variable setting $g(z)=f(e^z)$ then $g(z+a)=f(ke^z)$ where $k=e^a$. Now you have a linear difference equation. Difference equations have been well studied. A general method is some kind of Fourier transform, look for a solution in the form of exponential, then take a sum of those. See, for example, On equation f(z+1)-f(z)=f'(z), or the book of Gelfond, Calculus of finite differences.

In your case, set $f(z)=az^2$. This will satisfy the equation if $\sum C_jk_j^2\neq 0$.