A study of the q-analogue of the paranormed Cesàro sequence spaces

In this article, we introduce and investigate the q-Cesaro matrix C(q)=(cquv) with q is an element of(0,1) for which we have c(uv)(q)= {q(v)/[u+1](q )(0 <= v <= u) 0 (v>u)where the q-number [kappa](q)is given, as usual in theq-theory, by[kappa]q:={1-q(kappa)/1-q (kappa is an element of C)(n-1)Sigma P(k=0)q(k)=1+q+q2+<middle dot><middle dot><middle dot>+q(n-1) (kappa=n is an element of N), CandNbeing the sets of complex numbers and positive integers, respectively. Theq-Ces`aro matrixC(q) isaq-analogue of the Ces`aro matrixC1. We study the sequence spacesXq(p),Xq0(p),Xqc(p) andXq infinity(p), which areobtained by the domain of the matrixC(q) in the Maddox spaces & ell;(p),c0(p),c(p) and & ell;infinity(p),respectively. Wederive the Schauder basis and the alpha-, beta- and gamma-duals of these newly-defined spaces. Moreover,we state and prove several theorems characterizing matrix transformation from the spacesXq(p),Xq0(p),Xqc(p)andXq infinity(p) to anyone of the spacesc0,cor & ell;infinity