A Note on Bivariate bi-periodic Mersenne Polynomials

We introduce a new generalization mn(x,y), which we will call Bivariate Bi-periodic Mersenne polynomials depending on whether n is even or odd. We investigated the bivariate and biperiodic forms of Mersenne polynomials, focusing on their unique structural properties, roots, and relationships among coefficients. Key contributions include deriving the generating function and Binet formula, examining the limit behavior of the polynomials, and identifying connections between positive and negative terms. Also, we give the limits of the consecutive terms of the polynomials and some important identities such as Catalan, Cassini and D’Ocagne’s identity. We also find the corresponding binomial addition formula.