Analytical properties of 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. 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.The transform is closely related to both the q-deformed Poisson probability distribution, which is used widely in the q-boson operator calculus, and to Valiron's method of summation for divergent series. In general, Λqf is a meromorphic function whose poles are contained in the set Jq:={−q−j}j=0∞.In this paper, we study the connection between the behaviour of f on [0,1] and the decay of Λqf as z→∞.