# General four-term recurrence relations

I was perusing through this paper and I was wondering if there is any literature on general four term recurrence relations. Specifically, say a recurrence of the form $$A(n)u_{n+3}+B(n)u_{n+2}+C(n)u_{n+1}-u_n=0$$

where $A,B$ and $C$ are polynomials in $n$. I'm just curious because I've noticed in my research that recurrences of this form usually "pop up" when trying to find an analytic solution of certain differential equations, and it is ultimately is linked to whether the differential equation has a closed form solution of not. Also, does anyone know whether there's a theorem which gives conditions under which a recurrence does not admit any solutions?

## 1 Answer

You should look at Marko Petkovsek's thesis, and Petkovsek-Zeilberger's A=B (Petkovsek's thesis: Petkovšek, Marko. "Hypergeometric solutions of linear recurrences with polynomial coefficients." Journal of symbolic computation 14.2 (1992): 243-264.)

• You mean Wilf--Zeilberger, right? – Boris Bukh Mar 6 '15 at 1:49
• I do mean Petkovsek-Zeilberger's book... – Igor Rivin Mar 6 '15 at 4:27
• Aren't we both wrong: according to math.upenn.edu/~wilf/AeqB.html the authors are Petkovsek, Wilf, Zeilberger. – Boris Bukh Mar 6 '15 at 14:00
• @BorisBukh Yes, I had always thought of the book as Petkovsek-Zeilberger (always = 20 years), but apparently was wrong all this time... – Igor Rivin Mar 6 '15 at 14:39