Analytical properties of the Lupaş -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 ( Λ q f ) ( z ) : = 1 ( − z ; q ) ∞ ⋅ ∑ k = 0 ∞ f ( 1 − q k ) q k ( k − 1 ) / 2 ( q ; q ) k z k . ansform 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, Λ q f is a meromorphic function whose poles are contained in the set J q : = { − q − j } j = 0 ∞ . s paper, we study the connection between the behaviour of f on [ 0 , 1 ] and the decay of Λ q f as z → ∞ .