The existential closedness problem for a function f is to show that a system of complex polynomials in 2n variables always has solutions in the graph of f, except when there is some geometric obstruction. Special cases have be proven for exp, Weierstrass p functions, the Klein j function, and other important functions in arithmetic geometry using a variety of techniques. Recently, some special cases have also been studied for well-known solutions of difference equations using different methods. There is potential to expand on these results by adapting the strategies used to prove existential closedness results for functions in arithmetic geometry to work for analytic solutions of difference equations.