Linear recurrence with polynomial coefficients

[Pere Callahan]

Hi all,

I came across a linear recurrence with polynomial coefficients and realized that I don't have a clue as to how to solve it. The usual methods like generating functions or guessing seem not to work in that case.

Here is the equation:

$$i (i - 1) (i - 2) b<i> = 1/3 (i + 1) i (1 - i) b[i - 3] + (1 + i) i (i - 2) b[i - 2] + (i + 1) (i - 1) (i - 2) b[i - 1]<br /> </i>$$


Is there any general theory on recursions of that type or maybe even a general algorithm to compute the solution (in terms of some initial valeus b[0], b[1], b[2])?

Thanks a lot!

Pere

 

[Pere Callahan]

So, for this particular recursion, the substitution $b<i>\mapsto(i+1)c</i>$ reduces the equation to a very easy form. The question about the general theory remains.