# Two variable recurrence relations

I'm interested in solving the following type of problem...

Starting with a recurrence relation in multiple variables, for example:

$$v_{i-1,j} + v_{i,j+1} + v_{i+1,j} + v_{i,j-1} = 4v_{i,j}$$

with some boundary conditions, e.g.

$$v_{-1,j} = v_{i,-1} = v_{N,i} = v_{j,N} = 0$$

I wish to find the solution to the series

$$s_j = \sum_i v_{i,j}$$

(the terms of the relation can be more general, e.g. $kiv_{i+1,j-1}$)

How would you go about solving the relationship. It seems that often, simple generating functions can lead to differential equations that are difficult to solve. The use of $s$ seems like it will afford some tricks, but I'm struggling a bit here.

So, how would you go about finding the solution to this, and problems like it? Is there some general solutions to try? Will solutions be particular hypergeometric functions? If so, is there a way of narrowing down the possibilities to guess a solution?

Any hints or tips will be useful.

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If the differential equation you are starting from is hard to solve, the difference equation will be hopeless. – vonbrand Feb 6 '14 at 15:39