 What are some of interesting results that arise from using difference equations in number theory , Combinatorics or any other field ?



Hrushovski used the model theory of difference fields to give another proof of the ManinMumford conjecture. 


In analysis over fields of positive characteristic, the role of differential operators is played by special difference operators (the Carlitz derivative and its generalizations). In particular, the main special functions of that theory satisfy some difference equations. For the details see my book "Analysis in Positive Characteristic" (Cambridge University Press, 2009). 


The threeterm recurrence relation satisfied by a family of orthogonal polynomials is a crucial fact which brings together classical analysis, spectral theory and other branches of mathematics. This recurrence relation is obviously an example of a difference equation. 


In signal processing, digital filtering, ARMA modelesation, use difference equations. In Linear prediction Coding, we seek a difference operator such that: $x(n)=\sum_p{a(n)x(np)}+e(n)$ where $x(n)$ is for example a time serie (the uknown function in the difference equation). The goal of this coding is to find $a(n)$ by minimizing the error $e(n)$ so this is an example of an inverse problem for the difference operator given by the above equation. Like diffrential operators which are diagonalized by the Fourier transform, difference operators are diagonalized by the ZTransform defined by: $X(Z)= \sum_n{x(n)Z^{n}}$ where $Z$ is a complex variable lying on the unit circle of the complex plane ($Z=\exp(if)$). So a difference operator is transformed via the Ztransform to a multiplication operator by a polynom $P(Z)=\sum_p{a(p)Z^{p}}$ and one method for the solution of the problem is the use of orthogonal polynoms over the unit circle. 

