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On the polynomial solution of divided-difference equations of the hypergeometric type on nonuniform lattices. (English) Zbl 1432.33016

Summary: In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, difference or \(q\)-difference equation of hypergeometric type. This is achieved by studying the properties of the mean operator and the divided-difference operator as well as by defining explicitly, the right and the “left” inverse for the second operator. The method constructed to provide this formal proof is likely to play an important role in the characterization of orthogonal polynomials on non-uniform lattices and might also be used to provide hypergeometric representation (when it does exist) of the second solution – non polynomial solution – of a second-order divided-difference equation of hypergeometric type.

MSC:

33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
39A13 Difference equations, scaling (\(q\)-differences)
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