## On the complex difference equation of hypergeometric type on non-uniform lattices.(English)Zbl 1440.39007

Summary: 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.

### MSC:

 39A45 Difference equations in the complex domain 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 33E30 Other functions coming from differential, difference and integral equations 33E50 Special functions in characteristic $$p$$ (gamma functions, etc.)
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