Ostrovska, SofiyaOstrovska, SofiyaMathematics2024-07-052024-07-05201250022-247X1096-081310.1016/j.jmaa.2012.04.0472-s2.0-84861676281https://doi.org/10.1016/j.jmaa.2012.04.047https://hdl.handle.net/20.500.14411/322The Lupas q-transform emerges in the study of the limit q-Lupas operator. The latter comes out naturally as a limit for a sequence of the Lupas q-analogues of the Bernstein operator. Given q is an element of (0, 1), f is an element of C left perpendicular0, 1right perpendicular, the q-Lupas transform off is defined by (Lambda(q)f) (z) := 1/(-z; q)(infinity) . Sigma(infinity)(k=0) f(1 - q(k))q(k(k -1)/2)/(q; q)(k)z(k). 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, Lambda(q)f is a meromorphic function whose poles are contained in the set J(q) := {-q(-j)}(j=0)(infinity). In this paper, we study the connection between the behaviour of f on leftperpendicular0, 1right perpendicular and the decay of Lambda(q)f as z -> infinity. (C) 2012 Elsevier Inc. All rights reserved.eninfo:eu-repo/semantics/openAccessq-integersq-binomial theoremLupas q-analogue of the Bernstein operatorLupas q-transformAnalytic functionMeromorphic functionArticleQ2Q23941177185WOS:000305312500015