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

TL;DR

This paper provides explicit, exact solutions to a specific finite-difference equation, including applications such as matrix diagonalization, isospectral operators, and black hole problems, with implementations in Mathematica.

Contribution

The paper introduces a complete explicit solution to a nontrivial finite-difference equation, along with applications and implementations in Mathematica, which is novel in providing exact formulas and practical uses.

Findings

Explicit solutions to the finite-difference equation are derived.

Applications include matrix diagonalization and isospectral operators.

Discretised Laguerre polynomials are explicitly defined.

Abstract

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.