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Hahn Difference Operator and Associated Jackson–Nörlund Integrals

Author

Listed:
  • M. H. Annaby

    (Qatar University)

  • A. E. Hamza

    (Cairo University)

  • K. A. Aldwoah

    (Jazan University)

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.

Suggested Citation

  • M. H. Annaby & A. E. Hamza & K. A. Aldwoah, 2012. "Hahn Difference Operator and Associated Jackson–Nörlund Integrals," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 133-153, July.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:1:d:10.1007_s10957-012-9987-7
    DOI: 10.1007/s10957-012-9987-7
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    References listed on IDEAS

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    1. A. B. Malinowska & D. F. M. Torres, 2010. "The Hahn Quantum Variational Calculus," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 419-442, December.
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    Cited by:

    1. Aglić Aljinović, Andrea & Kovačević, Domagoj & Puljiz, Mate & Žgaljić Keko, Ana, 2021. "On Ostrowski inequality for quantum calculus," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    2. Cardoso, J.L. & Shehata, Enas M., 2024. "Hermite-Hadamard inequalities for quantum integrals: A unified approach," Applied Mathematics and Computation, Elsevier, vol. 463(C).
    3. Nichaphat Patanarapeelert & Thanin Sitthiwirattham, 2019. "On Fractional Symmetric Hahn Calculus," Mathematics, MDPI, vol. 7(10), pages 1-18, September.

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    1. Nichaphat Patanarapeelert & Thanin Sitthiwirattham, 2019. "On Fractional Symmetric Hahn Calculus," Mathematics, MDPI, vol. 7(10), pages 1-18, September.

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