Abstract. The spaces $\Delta c_{0}(p),$ $\Delta c(p) $ and $\Delta\ell _{\infty }(p) $ were defined by Ahmad and Mursaleen [1]. In [2], Altay and Polat also defined the sequence spaces $e_{0}^r(\Delta ) $, $e_{c}^r(\Delta )$ and $e_{\infty }^r(\Delta )$, and determined the absolute Köthe--Toeplitz duals of these spaces. The main purpose of this work is to introduce some new paranormed sequence spaces defined by Euler and difference operators and to compute their $\alpha $-, $\beta $-, $\gamma $-duals. Also some of the matrix transformations has been characterized.

AMS Subject Classification (1991): 46A45, 46B45

Keyword(s): paranormed sequence space, matrix mapping, Köthe--Toeplitz duals, Euler and difference sequence spaces

Received October 18, 2008, and in revised form June 9, 2009. (Registered under 57/2010.)