Insights into (?,?)-quantum difference relative uniform convergence

Abstract

This paper presents a new sequence space of functions, denoted as m lambda(Pi,ru,del(q,p)), which is defined via (lambda,mu)-relative uniform convergence and the Jackson q-difference operator. The space combines aspects of (lambda,mu)-convergence, relative uniform convergence, and quantum deformation, shedding light on the interaction between q-calculus and (lambda,mu)-averaging. We establish that m lambda(Pi,ru,del(q,p)) is a Banach space and satisfies the properties of both BK-spaces and K-spaces when equipped with a q-deformed norm. Further, we explore the geometric properties of this space, including strict convexity, uniform convexity, and the Opial-type condition, all of which depend on the parameters q and (lambda,mu). A q-dependent modulus of convexity is formulated, and we establish inclusion relations with classical spaces, such as l(p) spaces and (lambda, mu) difference spaces, including the limiting case as q -> 1. These results enrich the theory of sequence spaces and present a unified framework that bridges (lambda,mu)-convergence, relative uniform convergence, and quantum calculus.