On the Complex Difference Equation of Hypergeometric Type on Non-uniform Lattices

In this article, we obtain a new fundamental theorems for Nikiforov-Uvarov-Suslov complex difference equation of hypergeometric type by the method of Euler integral transformation, its expression is different from Suslov's Theorem. We also 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 and other important results.