On the Lupaş q-transform

The Lupaş q-transform emerges in the study of the limit q-Lupaş operator. The latter comes out naturally as a limit for a sequence of the Lupaş q-analogues of the Bernstein operator. Lately, it has been studied by several authors from different perspectives in mathematical analysis and approximation theory. This operator is closely related to the q-deformed Poisson probability distribution, which is used widely in the q-boson operator calculus. Given q ∈ (0, 1), f ∈ C[0, 1], the q-Lupaş transform of f is defined by: (Λqf )(z) := 1 (−z; q)∞ · ∞− k=0 f (1 − qk)qk(k−1)/2 (q; q)k zk. In this paper, we study some analytic properties of (Λqf )(z). In particular, we examine the conditions under which Λqf can either be an entire function, or a rational one.