Global Journal of Science Frontier Research, F: Mathematics and Decision Science, Volume 21 Issue 4

© 2021 Global Journals 1 Global Journal of Science Frontier Research Volume XXI Issue IV Year 2021 28 ( F ) Version I Difference Sequence Spaces of Second order Defined by a Sequence of Moduli ` ∞ ( 4 ) = { x = ( x k ) ∈ ω : ( 4 x k ) ∈ ` ∞ } c ( 4 ) = { x = ( x k ) ∈ ω : ( 4 x k ) ∈ c } c 0 ( 4 ) = { x = ( x k ) ∈ ω : ( 4 x k ) ∈ c 0 } where 4 x = ( x k − x k +1 ). These are Banach spaces with the norm || x || 4 = | x 1 | + ||4 x || ∞ . Mikail [14] defined the sequence spaces ` ∞ ( 4 2 ) = { x = ( x k ) ∈ ω : ( 4 2 x k ) ∈ ` ∞ } c ( 4 2 ) = { x = ( x k ) ∈ ω : ( 4 2 x k ) ∈ c } c 0 ( 4 2 ) = { x = ( x k ) ∈ ω : ( 4 2 x k ) ∈ c 0 } Where ( 4 2 x ) = ( 4 2 x k ) = ( 4 x k − 4 x k +1 ) . The sequence spaces ` ∞ ( 4 2 ) , c ( 4 2 ) and c 0 ( 4 2 ) are Banach spaces with the norm || x || 4 = | x 1 | + | x 2 | + ||4 2 x || ∞ . Mikail and Colak [15] defined the sequence spaces ` ∞ ( 4 m ) = { x = ( x k ) ∈ ω : ( 4 m x k ) ∈ ` ∞ } c ( 4 m ) = { x = ( x k ) ∈ ω : ( 4 m x k ) ∈ c } c 0 ( 4 m ) = { x = ( x k ) ∈ ω : ( 4 m x k ) ∈ c 0 } where m ∈ N , 4 0 x = ( x k ) , 4 x = ( x k − x k +1 ) , 4 m x = ( 4 m − 1 x k − 4 m − 1 x k +1 ) , and so that 4 m x k = m X i =0 ( − 1) i m i x k + i . 14. Mikail. On some difference sequence spaces, Dogra-Tr:J:Math., 17,(1993)18- 24. R ef