On the Lupaş q-transform

AbstractThe 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)kzk. 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