On k−division Fibonacci-Pell and k−division Gaussian Fibonacci-Pell polynomials and their applications

Creative Commons License

Mehraban E., Kuloğlu B., Özkan E., Hıncal E.

JOURNAL OF COMBINATORIAL MATHEMATICS AND COMBINATORIAL COMPUTING, cilt.128, ss.317-335, 2026 (Scopus)

Özet

This paper introduces two novel sequences: the \(k-\)-division Fibonacci--Pell polynomials and the \(k-\)-division Gaussian Fibonacci--Pell polynomials. Building on the well-known Fibonacci and Pell sequences, these new sequences are defined using a division-based approach, enhancing their combinatorial and algebraic properties. We present explicit recurrence relations, generating functions, combinatorial identities, and Binet-type formulas for these sequences. A significant contribution of the study is the factorization of the Pascal matrix via the Riordan group method using the proposed polynomials. Two distinct factorizations are derived, highlighting the algebraic structure and combinatorial interpretations of the \(k-\)-division polynomials. The work not only generalizes known polynomial sequences but also provides new insights into their matrix representations and applications.