METHODS AND APPLICATIONS OF ANALYSIS. c  2015 International Press Vol. 22, No. 4, pp. 343–358, December 2015 001 VARIATIONS AROUND JACKSON’S QUANTUM OPERATOR ∗ J. L. CARDOSO † AND J. PETRONILHO ‡ Abstract. Let 0 <q< 1, ω ≥ 0, ω 0 := ω/(1 - q), and I a set of real numbers. Consider the so-called quantum derivative operator, Dq,ω, acting on functions f : I → K (where K = R or C) as Dq,ω[f ](x) := f ( qx + ω ) - f (x) (q - 1)x + ω , x ∈ I \{ω 0 } , and Dq,ω[f ](ω 0 ) := f ′ (ω 0 ) whenever ω 0 ∈ I and this derivative exists. This operator was intro- duced by W. Hahn in 1949. Its inverse operator is given in terms of the so-called Jackson-Thomae (q, ω)-integral, also called Jackson-N¨ orlund (q, ω)-integral. For ω = 0 one obtains the Jackson’s q-operator, Dq , whose inverse operator is given in terms of the so-called Jackson q-integral. In this paper we survey in an unified way most of the useful properties of the Jackson’s q-integral and then, by establishing links between Dq,ω and Dq , as well as between the q and the (q, ω) integrals, we show how to obtain the properties of Dq,ω and the (q,ω)-integral from the corresponding ones fulfilled by Dq and the q-integral. We also consider (q, ω)-analogues of the Lebesgue spaces, denoted by L p q,ω [a, b] and L p q,ω [a, b], being a, b ∈ R. It is shown that the condition a ≤ ω 0 ≤ b ensures that these are indeed linear spaces over K. Moreover, endowed with an appropriate norm, L p q,ω [a, b] satisfies some expected properties: it is a Banach space if 1 ≤ p ≤∞, separable if 1 ≤ p< ∞, and reflexive if 1 <p< ∞. Key words. Jackson q-integral, Jackson-N¨ orlund (q,ω)-integral, (q, ω)-Lebesgue spaces, q-analogues. AMS subject classifications. 33E20, 33E30, 40A05, 40A10. 1. Introduction. Quantum calculus is, roughly speaking, the equivalent to tra- ditional infinitesimal calculus without the notion of limits. In quantum calculus one defines the so called Jackson q−derivative, D q [f ](x) := f (qx) − f (x) (q − 1)x , and the (forward difference) ω−derivative, △ ω [f ](x) := f (x + ω) − f (x) ω , and they can be treated together using the more general (q,ω)−derivative operator, D q,ω [f ](x) := f ( qx + ω ) − f (x) (q − 1)x + ω , often called the Hahn’s quantum operator. (In order to simplify this introduction we omit some details related with these definitions.) The subject is old (D q,ω was introduced by W. Hahn in 1949) but is receiving an increasing interest, e.g., in ap- plications to Physics, Approximation Theory, and Optimization—see for instance the works [3, 7, 8, 9, 10, 12, 15, 16, 17, 19, 22, 23, 24, 25, 28], and references therein. ∗ Received September 10, 2013; accepted for publication April 24, 2015. † Departamento de Matem´ atica, Escola das Ciˆencias e Tecnologia, Universidade de Tr´as-os-Montes e Alto Douro, UTAD, Quinta de Prados, 5001-801 Vila Real, Portugal (jluis@utad.pt). ‡ CMUC and Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal (josep@mat.uc.pt). 343