In this research paper, we undertake an investigation into Cesàro \(\mathfrak {q}\)-difference sequence spaces \(\mathfrak {X}(\mathfrak {C}_1^{\delta ;\mathfrak {q}})\), where \(\mathfrak {X} \in \{\ell _{\infty },c,c_0\}.\) These spaces are generated using the matrix \(\mathfrak {C}_1^{\delta ,\mathfrak {q}}\), which is a product of the Cesàro matrix \(\mathfrak {C}_1\) of the first-order and the second-order \(\mathfrak {q}\)-difference operator \(\nabla ^2_\mathfrak {q}\) defined by

where \(\mathfrak {q}\in (0,1)\) and \(\mathfrak {f}_k=0\) for \(k<0.\) Our endeavor includes the establishment of significant inclusion relationships, the determination of bases for these spaces, the investigation of their \(\alpha \)-, \(\beta \)-, and \(\gamma \)-duals, and the formulation of characterization results pertaining to matrix classes \((\mathfrak {X},\mathfrak {Y})\), with \(\mathfrak {X}\) chosen from the set \(\{\ell _{\infty }(\mathfrak {C}_1^{\delta ;\mathfrak {q}}), c(\mathfrak {C_1^{\delta ;\mathfrak {q}}}), c_0(\mathfrak {C}_1^{\delta ;\mathfrak {q}})\}\) and \(\mathfrak {Y}\) chosen from the set \(\{\ell _{\infty },c,c_0,\ell _{1}\}.\) The final section of our study is dedicated to the meticulous spectral analysis of the weighted \(\mathfrak {q}\)-difference operator \(\nabla ^{2;\mathfrak {z}}_{\mathfrak {q}}\) over the space \(c_0\) of null sequences.

在这篇研究论文中，我们对 Cesàro \(\mathfrak {q}\)差分序列空间\(\mathfrak {X}(\mathfrak {C}_1^{\delta ;\mathfrak {q}})进行了研究\)，其中\(\mathfrak {X} \in \{\ell _{\infty },c,c_0\}.\)这些空间是使用矩阵\(\mathfrak {C}_1^{\delta ,\mathfrak {q}}\) ，它是一阶和二阶\(\mathfrak {q}\)差分算子的 Cesàro 矩阵\(\mathfrak {C}_1\)的乘积\(\nabla ^2_\mathfrak {q}\)定义为

其中\(\mathfrak {q}\in (0,1)\)和\(\mathfrak {f}_k=0\)对于\(k<0.\)我们的努力包括建立重要的包含关系，确定这些空间的基，研究它们的\(\alpha \) -、\(\beta \) - 和\(\gamma \) -对偶，并制定与矩阵类相关的表征结果\(( \mathfrak {X},\mathfrak {Y})\)，其中\(\mathfrak {X}\)从集合\(\{\ell _{\infty }(\mathfrak {C}_1^{\)中选择delta ;\mathfrak {q}}), c(\mathfrak {C_1^{\delta ;\mathfrak {q}}}), c_0(\mathfrak {C}_1^{\delta ;\mathfrak {q}}) \}\)和\(\mathfrak {Y}\)从集合\(\{\ell _{\infty },c,c_0,\ell _{1}\}.\)中选择。研究致力于对空间上的加权\(\mathfrak {q}\)差分算子\(\nabla ^{2;\mathfrak {z}}_{\mathfrak {q}}\)进行细致的谱分析\(c_0\)空序列。