ON SEQUENCE SPACES GENERATED BY BINOMIAL DIFFERENCE OPERATOR OF FRACTIONAL ORDER

In this article we introduce binomial di ff erence sequence spaces of fractional order alpha; b(p)(r,s) (Delta((alpha))) (1 <= p <= infinity) by the composition of binomial matrix, B-r,B-s and fractional di ff erence operator Delta((alpha)), defined by (Delta((alpha))x)(k) = Sigma(infinity)(i=0) (-1)(i) Gamma(alpha+1)/i!Gamma(alpha-i+1)x(k-i). We give some topological properties, obtain the Schauder basis and determine the alpha, beta and gamma-duals of the spaces. We characterize the matrix classes b(p)(r,s) (Delta((alpha))), Y); where Y is an element of {l(infinity), c, c(0,) l(1)} and certain classes of compact operators on the space b(p)(r,s) (Delta((alpha))) using Hausdorff measure of non-compactness. Finally, we give some geometric properties of the space b(p)(r,s) (Delta((alpha))) (1 < p < infinity). (C) 2019 Mathematical Institute Slovak Academy of Sciences