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Hahn Difference Operator and Associated Jackson–Nörlund Integrals

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Physics, Qatar University, P.O. Box. 2713, Doha, Qatar

    M. H. Annaby

  2. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

    A. E. Hamza

  3. Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia

    K. A. Aldwoah

Authors
  1. M. H. Annaby
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  2. A. E. Hamza
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  3. K. A. Aldwoah
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Correspondence to M. H. Annaby.

Additional information

M.H. Annaby is on leave from: Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.

K.A. Aldwoah is on leave from: Department of Mathematics, Faculty of Education and Applied Science, Hajjah University, Hajjah, Yemen.

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Annaby, M.H., Hamza, A.E. & Aldwoah, K.A. Hahn Difference Operator and Associated Jackson–Nörlund Integrals. J Optim Theory Appl 154, 133–153 (2012). https://doi.org/10.1007/s10957-012-9987-7

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  • DOI: https://doi.org/10.1007/s10957-012-9987-7

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