Hahn Difference Operator and Associated Jackson–Nörlund Integrals

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.Keywords Hahn's operator · Difference equations · q-Difference equations · Nörlund sums · Jackson q-integral M.H. Annaby is on leave from:

The Hahn Quantum Variational Calculus

“…The Hahn difference operator has the following properties: 17,18]). (a) Let f be q, ω-differentiable on I and D q,ω f ≡ 0 on I.…”

Existence Results for Fractional Hahn Difference and Fractional Hahn Integral Boundary Value Problems

“…The right inverse of the Hahn difference operator was proposed by Aldwoah in 2009 [14,15]. This operator is in the terms of the Jackson -integral containing the right inverse of [16] and Nörlund sum containing the right inverse of Δ [16].…”

Some results on quantum Hahn integral inequalities

“…[3][4][5][6]). For some recent results to boundary value problems for Hahn difference operators, we refer to papers [7][8][9][10] and the references cited therein. Let [a, b] ⊆ R be an interval.…”