Exact, explicit and entire solutions to a nontrivial finite-difference equation and their applications

Below, the explicit solution to a certain finite-difference equation is given and the required steps for derivation of these results are outlined. Everything is included as Mathematica formulae, so the notebook itself can be used for checking and improving the present results. Some important references for justifying some steps and crosschecking certain results have been included. Full references and derivations will be made available shortly. It should be noted that several applications for the solutions have been included at the end of the document. These include at least diagonalisation of certain infinite matrices, definition of isospectral operators with simple eigenvalues and alternative representation of a solution to a problem related to black holes. Additionally, the corresponding, discretised associated Laguerre polynomials are defined explicitly.