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.

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