Kizmaz [13] studied the difference sequence spaces $\ell_\infty(\Delta),~c(\Delta)$, and $c_0(\Delta)$. Several article dealt with the sets of sequences of $m$-th order difference of which are bounded, convergent, or convergent to zero. Altay and Ba\c sar [5] and Altay, Ba\c sar, and Mursaleen [7] introduced the Euler sequence spaces $e_0^r$, $e_c^r$, and $e_\infty^r$, respectively. The main purpose of this article is to introduce the spaces $e_0^r(\Delta^{(m)})$, $e_c^r(\Delta^{(m)})$, and $e_\infty^r(\Delta^{(m)})$ consisting of all sequences whose $m^{th}$ order differences are in the Euler spaces $e_0^r$, $e_c^r$, and $e_\infty^r$, respectively. Moreover, the authors give some topological properties and inclusion relations, and determine the $\alpha$-, $\beta$-, and $\gamma$-duals of the spaces $e_0^r(\Delta^{(m)})$, $e_c^r(\Delta^{(m)})$, and $e_\infty^r(\Delta^{(m)})$, and the Schauder basis of the spaces $e_0^r(\Delta^{(m)})$, $e_c^r(\Delta^{(m)})$. The last section of the article is devoted to the characterization of some matrix mappings on the sequence space $e_c^r(\Delta^{(m)})$.
