Extending Mersenne Numbers to Bicomplex (p,q)-Mersenne Quaternions with Catalan Transformations

In our study we define bicomplex (p,q)- Mersenne numbers. Utilizing these numbers, we present bicomplex (p,q)- Mersenne quaternions which are a generalization of Mersenne quaternions. All these sequences have second order recurrence relations. We obtain Binet formula, the generating functions, the exponential generating function, Catalan identity, Cassini identity, D’ocagne’s identity, summation formula for both. We also introduce matrix form of bicomplex (p,q)- Mersenne quaternions. Also, we introduce a novel generalization of Mersenne numbers by incorporating Catalan numbers into the quaternionic (p,q)- Mersenne sequence. By extending Mersenne numbers to a quaternionic framework, we establish a new connection between Catalan numbers and quaternion structures. This approach provides deeper insights into the algebraic and combinatorial properties of these generalized sequences. Potential future research directions include further exploration of their structural characteristics and applications in number theory and related fields. Also, We introduced the quaternion-type Catalan transform for Mersenne numbers, demonstrating how it amplifies growth from exponential to super-exponential and faster-than-quadratic rates. This transformation enhances the sequence’s structural complexity and information density.