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Does anyone know of a paper that may have been written on $3^{rd}$ order recurrence relations with polynomial coefficients, that is, one of the form

$$A(n)a_{n+3}+B(n)a_{n+2}+C(n)a_{n+1}=D(n)a_n$$

I've come across one in my research and it appears to be quite intractable.

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1 Answer 1

A few cases are tractable, if e.g. the recurrence takes the form: $$ n^{\overline{k}} \Delta^k a_n + c_{k - 1} n^{\overline{k - 1}} \Delta^{k - 1} a_n + \ldots + c_0 a_n = 0 $$ it is a Cauchy-Euler difference equation, which has a solution of the form $a_n = n^{\overline{r}}$. In the above $n^{\overline{k}} = n (n + 1) \ldots (n + k - 1)$ is a rising factorial power, $\Delta a_n = a_{n + 1} - a_n$ is the difference operator, and the $c_i$ are constants. Details in Wikipedia, but with ghastly notation.

If you can guess a solution, reduction of order (just like for differential equations) might help.

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