Recursions are quite important mathematical tools since many systems are mathematically modelled to ultimately take us to these equations because of their rather easy algebraic natures. They fit computer programming needs quite well in many circumstances to produce solutions. However, it is generally desired to find the asymptotic behaviour of the general term in the relevant sequence for convergence and therefore practicality issues. One of the general tendencies to find the general term asymptotic behaviour, when its ordering number grows unboundedly, is the integral representation over a generating function which does not depend on individual sequence elements. This is tried to be done almost for all types of recursions, even though the linear cases gain more importance than the others because they can be more effectively investigated by using many linear algebraic tools. Despite this may seem somehow to be rather trivial, there are a lot of theoretical fine tuning issues in the construction of true integral representations over true intervals on real axis or paths in complex domains. This work is devoted to focus on this issue starting from scratch for better understanding of the matter. The example cases are chosen to best illuminate the situations to get information for future generalization even though the work can be considered at somehow introductory level.