Some Pascal Spaces of Difference Sequences Spaces of Order m

Abstract

The main purpose of this article is to introduce new sequence spaces $p_{\infty }\left( \Delta ^{(m)}\right) $, $p_{c}\left( \Delta ^{(m)}\right) $ and $p_{0}\left( \Delta ^{(m)}\right) $ which are consisted by sequences whose $m^{th}$ order differences are in the Pascal sequence spaces $p_{\infty }$, $p_{c}$ and $p_{0}$, respectively. Furthermore, the bases of the new difference sequence spaces $p_{c}\left( \Delta ^{(m)}\right) $ and $p_{0}\left( \Delta ^{(m)}\right) ,$ and the $% \alpha $-, $\beta $-$\ $and $\gamma $-duals of the new difference sequence spaces $p_{\infty }\left( \Delta ^{(m)}\right) $, $p_{c}\left( \Delta ^{(m)}\right) $ and $p_{0}\left( \Delta ^{(m)}\right) \ $have been determined. Finally, necessary and sufficient conditions on an infinite matrix belonging to the classes $(p_{c}\left( \Delta ^{(m)}\right) :l_{\infty })$ and $(p_{c}\left( \Delta ^{(m)}\right) :c)$ are obtained.

Keywords

• Pascal difference sequence spaces,difference operator of order m

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