GAUSSIAN FERMAT NUMBERS, POLYNOMIALS AND THEIR ASSOCIATED TRANSFORMS

In this paper, we introduced Gaussian Fermat numbers and polynomials. We provided the Binet formula, generating functions, and the exponential generating function for these numbers and polynomials. Additionally, we derived several identities for these polynomials, including the Cassini identity, Catalan identity, Vajda identity, Halton identity, Gelin-Cesaro identity, and D’Ocagne’s identity. We demonstrated that Gaussian Fermat numbers and polynomials can also be obtained through matrix representations and discussed key propositions based on the fact that the determinants of these matrix representations are constant. Furthermore, we explored the relationship between Gaussian Fermat numbers, polynomials, Mersenne numbers, and Jacobsthal numbers. We also presented the Catalan, Binomial, and Binomial of Catalan transformations of the Gaussian Fermat sequence and polynomials. Finally, we introduced the generating function for the Catalan transformation of the Gaussian Fermat numbers and polynomials.