Ideal Convergence of Quantum Difference Sequences of Bi-Complex Numbers

Abstract

We study second-order quantum difference sequence spaces over bi-complex numbers. We define $Z[\nabla_q^2, \|\cdot\|_{\mathbb{C}_2}]$ for $Z \in \{I^*_c, I^*_\theta, I^*_1, I^*_p, I^*_\infty\}$, using the second order $q$- difference operator $\nabla_q^2$ under the Euclidean norm. We examine their $BK$-space structure, symmetric property, inclusion relations, and isomorphisms with classical $ I-$ convergent bi-complex sequence spaces. A matrix representation of the operator is given, along with counterexamples, to demonstrate contradictions in specific inclusion cases.