A Study of the $q$-Analogue of the Paranormed Ces\`{a}ro Sequence Spaces

Abstract

In this article, we introduce and investigate
the $q$-Ces\`{a}ro matrix $C(q)=(c^q_{uv})$ with $q\in (0,1)$
for which we have
\[
c^q_{uv}= \left\{\begin{array}{ll}
\dfrac{q^v}{[u+1]_q} &\qquad (0\leqq v \leqq u) \\
\\
0 & \qquad (v>u),
\end{array}
\right.
\]
where the $q$-number $[\kappa]_q$ is given,
as usual in the $q$-theory, by
\begin{equation*}
\left[\kappa\right]_{q}:=\left\{
\begin{array}{ll}
\dfrac{1-q^{\kappa}}{1-q} &\qquad (\kappa\in \mathbb{C})\\
\\
\sum\limits_{k=0}^{n-1}q^{k}=1+q+q^2+\cdots+q^{n-1}
&\qquad (\kappa=n\in \mathbb{N}),
\end{array}
\right.
\end{equation*}
$\mathbb{C}$ and $\mathbb{N}$ being the sets of
complex numbers and positive integers, respectively.
The $q$-Ces\`{a}ro matrix $C(q)$ is a $q$-analogue of
the Ces\`{a}ro matrix $C_1$. We study the sequence spaces
$X^q(p),$ $X^q_0(p),$ $X^q_c(p)$ and $X^q_\infty(p)$,
which are obtained by the domain of the matrix $C(q)$
in the Maddox spaces $\ell{(p)},$ $c_0(p),$ $c(p)$ and
$\ell_{\infty}(p),$ respectively. We derive 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 spaces $X^q(p),$ $X^q_0(p),$ $X^q_c(p)$ and
$X^q_\infty(p)$ to anyone of the spaces
$c_0,$ $c$ or $\ell_{\infty}.$