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On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices

1
Department of Mathematics, Higher Teachers’ Training College, University of Yaounde 1, Yaounde, Cameroon
2
The African Institute for Mathematical Sciences, Limbe, Cameroon
3
Department of Mathematics, Higher Teachers’ Training College, University of Maroua, Maroua, Cameroon
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Axioms 2019, 8(2), 47; https://doi.org/10.3390/axioms8020047
Received: 31 January 2019 / Revised: 26 March 2019 / Accepted: 10 April 2019 / Published: 21 April 2019
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
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Abstract

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. View Full-Text
Keywords: second-order differential/difference/q-difference equation of hypergeometric type; non-uniform lattices; divided-difference equations; polynomial solution second-order differential/difference/q-difference equation of hypergeometric type; non-uniform lattices; divided-difference equations; polynomial solution
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Foupouagnigni, M.; Mboutngam, S. On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices. Axioms 2019, 8, 47.

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