Asian Journal of Probability and Statistics

In this paper, we introduce and develop the concept of generalized dual hyperbolic Pierre numbers, a novel class of number sequences that extends the structural framework of classical Pierre-type sequences through duality and hyperbolic transformations. This generalization offers a unified approach that encompasses both established and newly constructed numerical models. As distinguished special cases, we examine the dual hyperbolic Pierre numbers and their Lucas-type counterparts, emphasizing their algebraic relationships and unique structural features. Our study presents a comprehensive set of mathematical results, including closed-form identities, matrix representations, and recurrence relations that define the behavior of these sequences. We further derive Binet-type formulas for explicit term computation and construct generating functions that capture their combinatorial and analytical properties. Additionally, we explore Simson’s formulas and establish various summation identities that reveal deeper interconnections among sequence elements. This investigation contributes to the broader theory of Pierre-type sequences and offers new tools for research in discrete mathematics, algebraic structures, and computational number theory.