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:

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In this paper, we consider a strictly increasing continuous function β, and we present a general quantum difference operator D β which is defined to be. This operator yields the Hahn difference operator when β(t) = qt + ω, the Jackson q-difference operator when β(t) = qt, q ∈ (0, 1), ω > 0 are fixed real numbers and the forward difference operator when β(t) = t + ω, ω > 0.A calculus based on the operator D β and its inverse is established. MSC: 39A10; 39A13; 39A70; 47B39Keywords: quantum difference operator; quantum calculus; Hahn difference operator; Jackson q-difference operator

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Hahn introduced the difference operator D q,ω f (t) = (f (qt + ω) -f (t))/(t(q -1) + ω) in 1949, where 0 < q < 1 and ω > 0 are fixed real numbers. This operator extends the classical difference operator ω f (t) = (f (t + ω) -f (t))/ω as the Jackson q-differenceIn this paper, we present new results of the calculus based on the Hahn difference operator. Also, we establish an existence and uniqueness result of solutions of Hahn difference equations by using the method of successive approximations.