# Research in Scientific Computing in Undergraduate Education

### Existence of solutions to recurrence relations

There are many open problems in this area. Here is one from the book by Kulenovic and Ladas.

A solution to the recurrence relation $x_{n+2}=-1+\frac{x_n}{x_{n+1}}$ is a sequence of real numbers $a_1,a_2,a_3, \ldots$ such that $a_{n+2}=-1+\frac{a_n}{a_{n+1}}$ for all $n\geq1$.

The existence problem for the recurrence relation is this: for which values of $a,b$ is there a solution $a_1,a_2,a_3, \dots$ of the recurrence relation with $a_1=a,a_2=b$?

References:

Kulenovic, M.R.S. & Ladas, G. (2002) Dynamics of second order rational difference equations. Chapman & Hall/CRC.

Camouzis, E., DeVault, R. & Lada, G. (2001) On the recursive sequence $x_{n+1}=-1+\frac{x_{n-1}}{x_n}$. J. Diff Equations Applied, vol 7.