We define in a natural way the Bicomplex analog of the frames (bc-frames) in the setting of  Bicomplex infinite Hilbert spaces, and we characterize them in terms of their idempotent components. We also extend some classical results from frames theory to bc-frames and show that some of them do not remain valid for bc-frames in general. The construction of bc-frame operators and Weyl--Heisenberg bc-frames are also discussed.

In this paper, we first establish the Bicomplex version of Rouche's theorem. Also, a new approach is given to prove the maximum modulus principle for Bicomplex holomorphic functions. Our proof is based on the direct method and extends the result proved by Luna-Elizarraras et al. Finally, we generalize the Hurwitz's theorem to Bicomplex space.

Quantum calculus, which arises in the mathematical fields of combinatorics and special functions as well as in a number of areas, involving the study of fractals and multi−, fractal measures, and expressions for the entropy of chaotic dynamical systems, has attracted the attention of many researchers in recent years. In this paper, by virtue of some useful notations from q−, calculus, we define the q−, Fibonacci dual Bicomplex numbers and q−, Lucas dual Bicomplex numbers with a different perspective. Afterwards, we give the Binet formulas, binomial sums, exponential generating functions, Catalan identities, Cassini identities, d’, Ocagne identities and some algebraic properties for the q−, Fibonacci dual Bicomplex numbers and q−, Lucas dual Bicomplex numbers.

In this paper, we present the concept of D􀀀, topological dual of a BC􀀀, module X with a D􀀀, norm k: kD, X and we , nd D􀀀, topological duals of D􀀀, normed Bicomplex BC􀀀, modules l| p (BC) for 1 ,p < 1: We also investigate some fundamental topological properties of Bicomplex BC􀀀, modules l| p (BC) by de, ning some topological concepts such as solid space, monotone space, BK􀀀, space, symmetric space in the Bicomplex setting by using the D􀀀, norm j: j|: