Multivariate recurrence solve

by Praneetmek   Last Updated August 01, 2018 21:20 PM

I am currently working on a project and have reduced much of the problem to a single recurrence equation:

$$g(x,y)=1-\frac{1}{6}g(y,x-1)-\frac{1}{2}g(y,x)-\frac{1}{3}g(y+1,x-1)$$ $$g(0,a)=1$$ $$g(a,0)=0$$ When solving the problem out by hand, plugging in an ordered pair (m,n) results in m+n systems of equations which can be solved out (if you don't believe me try it). However, I'm looking for either a closed form for this equation or a way to program a recurrence table. I tried running it in mathematica but it doesn't work because mathematica needs values to recurse on, whereas the problem must be solved using a system of many equations. Any kinda solution would be helpful, whether it be using a program or done by hand. A closed form is preferable but if not possible, a recurrence table at least.

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