Function $f(x)$ such that $f(x-i)+f(x)=\frac{1}{x^2}$

Question

Help me find a function $f(x)$ such that

$$f(x-i)+f(x)=\frac{1}{x^2}$$

where $i$ is the imaginary unit.

Answer 1

Here is one solution, in terms of a "polygamma" function. http://en.wikipedia.org/wiki/Polygamma_function

Let $f(x) = \psi^{(1)}(ix)$ then we have $f(x-i)-f(x) = 1/x^2$ if my calculations are correct. This is based on the formula for the diagmma $$ \psi(x+1)-\psi(x) = \frac{1}{x}, $$ then differentiate to get $1/x^2$ and use $ix$ since your increment is $-i$.