# Adjoint Difference Equation for a Nikiforov-Uvarov-Suslov difference equation of hypergeometric type on Non-uniform Lattices

## TL;DR

This paper develops the adjoint difference equation for the Nikiforov-Uvarov-Suslov hypergeometric type equation on non-uniform lattices, providing new solutions and fundamental theorems that extend existing mathematical frameworks.

## Contribution

It introduces the adjoint equation for the Nikiforov-Uvarov-Suslov difference equation and derives new fundamental theorems, expanding the theoretical understanding of hypergeometric difference equations.

## Key findings

- Derived the adjoint difference equation for the hypergeometric type
- Obtained particular solutions for the adjoint and original equations
- Proved new fundamental theorems different from Suslov's

## Abstract

In this article, we establish the adjoint equation for Nikiforov-Uvarov-Suslov difference equation of hypergeometric type on non-uniform lattices, and prove it to be a difference equation of hypergeometric type on non-uniform lattices as well. The particular solutions of the adjoint equation are then obtained. As an appliction of these particular solutions, we use them to obtain the particular solutions for the original difference equation of hypergeometric type on non-uniform lattices. Finally, we prove another kind of fundamental theorems for Nikiforov-Uvarov-Suslov difference equation of hypergeometric type, which are essentially new results, its expression is different from Suslov's Theorem.