Difference equations – N terms recursion relation with the 3-term recursion relation

I have a recurrence relation of N-term of the form
$ sum_ {j = 0} ^ {N-1} a_ {j, i} ^ {(N)} b_ {i-j} = 0 $,
or $ a_ {j, i} ^ {(N)} $ are the coefficients and we take $ i $ up to a finite value. I want to reduce it into a 3-term recurrence relation by using the Gaussian elimination method using the formula
$ a_ {j, i} ^ {(k)} = a_ {j, i} ^ {(k + 1)} $ when $ i <k $ and
$ a_ {j, i} ^ {(k)} = a_ {j, i} ^ {(k + 1)} – frac {a_ {k, i} ^ {(k + 1)} a_ {j- 1, i-1} ^ {(k)}} {a_ {k-1, i-1} ^ {(k)}} $ other.
I tried to do it in MATHEMATICA but failed. Can any one help me with that? A recurrence relationship of 5 terms with a 3-term relationship will also help.