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Consider a function of two variables $f(x,y)$. I am interested in solving the recurrence relation

$$a \left(f(x,y-1)-f(x,y+1)\right)=b \left(f(x-1,y)-f(x+1,y)\right)$$

for a most general $f(x,y)$, where $a$ and $b$ are constants.

Unfortunately, I am not completely sure how to approach this problem. Do some techniques exist that would allow to tackle this task? What would be a fruitful way to proceed?

Kagaratsch
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  • Over what values are you trying to solve this? What are you base cases? – Simply Beautiful Art Jan 26 '20 at 01:58
  • @SimplyBeautifulArt I'm considering $x,y\in\mathbb{C}$ while the integer spaced recurrences given encode certain periodicities of the function. I imagine a general solution would involve free constants (or even functions if not sufficiently constrained), so that one could accommodate an arbitrary initial condition after the fact. – Kagaratsch Jan 26 '20 at 02:49

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