Hahn Difference Operator and Associated Jackson–Nörlund Integrals

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...

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Bibliographic Details
Published in:Journal of optimization theory and applications 2012-07, Vol.154 (1), p.133-153
Main Authors: Annaby, M. H., Hamza, A. E., Aldwoah, K. A.
Format: Article
Language:English
Subjects:
Applications of Mathematics
Calculus
Calculus of Variations and Optimal Control
Optimization
Construction
Difference equations
Engineering
Integrals
Inverse
Investigations
Mathematical analysis
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Operators
Optimization
Science
Theory of Computation
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Summary: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.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-012-9987-7