Insights into (λ,μ)-quantum difference relative uniform convergence

This paper presents a new sequence space of functions, denoted as [Formula: see text], which is defined via [Formula: see text]-relative uniform convergence and the Jackson [Formula: see text]-difference operator. The space combines aspects of [Formula: see text]-convergence, relative uniform convergence, and quantum deformation, shedding light on the interaction between [Formula: see text]-calculus and [Formula: see text]-averaging. We establish that [Formula: see text] is a Banach space and satisfies the properties of both [Formula: see text]-spaces and [Formula: see text]-spaces when equipped with a [Formula: see text]-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 [Formula: see text] and [Formula: see text]. A [Formula: see text]-dependent modulus of convexity is formulated, and we establish inclusion relations with classical spaces, such as [Formula: see text] spaces and [Formula: see text] difference spaces, including the limiting case as [Formula: see text]. These results enrich the theory of sequence spaces and present a unified framework that bridges [Formula: see text]-convergence, relative uniform convergence, and quantum calculus.