On Exploring the r−Stirling Fibonacci Numbers and Polynomials

Abstract

This paper introduces and investigates a novel class of combinatorial constructs termed the $r$-Stirling Fibonacci numbers and polynomials of the first and second kind. By integrating the exponential generating functions of classical Fibonacci numbers with those of the signed $r$-Stirling numbers, we establish an enriched algebraic framework that advances the theory of special numbers and polynomials. The study yields new identities - including horizontal generating functions, explicit formulas, and convolution relations - that extend classical combinatorial results. In addition, we define the $r$-Stirling Chebyshev polynomials of both kinds by employing hyperbolic functions and exponential techniques, thereby forging a functional link between Fibonacci-type and Stirling-type sequences. These results are rigorously validated through series expansion and the Cauchy product method. The theoretical contributions of this work highlight the interplay between combinatorics, algebra, and analysis, with potential applications in number theory, orthogonal polynomials, and symbolic computation.