DIMORPHIC TRIMAT SEQUENCES: ALGEBRAIC PROPERTIES AND DEGENERATE EXTENSIONS

Abstract

In this paper, we introduce and investigate a new class of Fermat-type integer sequences and polynomials, called the Trimat numbers and Trimat polynomials, inspired by the recurrence structure of the Tribonacci sequence. Although motivated by a third-order sequence, the Trimat sequence satisfies a second-order recurrence relation, leading to several elegant identities and structural properties. We establish Binet-type formulas, ordinary and exponential generating functions, and derive a comprehensive collection of classical identities including the Catalan, Cassini, d’Ocagne, Honsberger, and Vajda identities. In addition, we explore matrix representations of the Trimat sequence and its polynomials and analyze determinant invariances associated with these matrices. Transformative structures such as the Hankel, Catalan, and binomial transforms are applied to the Trimat sequence to further reveal its internal algebraic and combinatorial behavior. Motivated by recent developments on degenerate special numbers, we also introduce a new dimorphic type of Fermat numbers and establish their connections with the degenerate Stirling polynomials of the second kind. These results broaden the framework of Fermat-related number sequences and provide new perspectives for both theoretical and applied research.