Hahn Difference Operator and Associated Jackson–Nörlund Integrals

  • M. H. Annaby
  • A. E. Hamza
  • K. A. Aldwoah


This paper is devoted for a rigorous investigation of Hahn’s difference operator and the associated calculus. Hahn’s difference operator generalizes both the difference operator and Jackson’s q-difference operator. Unlike these two operators, the calculus associated with Hahn’s difference operator receives no attention. In particular, its right inverse has not been constructed before. We aim to establish a calculus of differences based on Hahn’s difference operator. We construct a right inverse of Hahn’s operator and study some of its properties. This inverse also generalizes both Nörlund sums and the Jackson q-integrals. We also define families of corresponding exponential and trigonometric functions which satisfy first and second order difference equations, respectively.


Hahn’s operator Difference equations q-Difference equations Nörlund sums Jackson q-integral 


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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics and PhysicsQatar UniversityDohaQatar
  2. 2.Department of Mathematics, Faculty of ScienceCairo UniversityGizaEgypt
  3. 3.Department of Mathematics, Faculty of ScienceJazan UniversityJazanSaudi Arabia

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