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\begin{document}

\title{Generalized Order Fibonacci and Lucas Polynomials and Hybrinomials}



\author[1]{Süleyman Aydınyüz}%
\author[2]{Gul Karadeniz Gozeri}%
\affil[1]{Pamukkale University}%
\affil[2]{Istanbul University}%


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  \date{May 21, 2024}


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\begin{abstract}
In this study, we first define generalized order Fibonacci and Lucas
polynomials. We show that by special choices one can obtain some known
sequences of polynomials and numbers such as order Pell polynomials,
order Jacobsthal polynomials, order Fibonacci and Lucas numbers and etc.
by using the definition of order Fibonacci and Lucas polynomials. Then
we consider hybrid numbers and polynomials whose importance is
increasing in mathematics, physics and engineering day by day. We
generalize the hybrid polynomials by moving them to the order. Hybrid
polynomials that are defined with this generalization are called order
Fibonacci and Lucas hybrinomials throughout this paper. We define the
generalized order Fibonacci and Lucas hybrinomials using generalized
order Fibonacci and Lucas polynomials. Besides this, we give the
recurrence relations of the generalized order Fibonacci and Lucas
hybrinomials. Also, we show that by special choices in this recurrence
relations one can obtain some known hybrid polynomials such as Horadam,
Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas
hybrinomials. Furthermore, we introduce the generating functions of
hybrinomials and give some important properties. Finally, we define the
matrix representations of the generalized order Fibonacci and Lucas
hybrinomials. For this purpose, we derive the matrices of and that play
similar role to the matrix for Fibonacci numbers. We show that by
special choices of the integers and , one can obtain matrix
representations of some known hybrinomials such as Pell, Jacobsthal
hybrinomials and etc.%
\end{abstract}%



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\textbf{Hosted file}

\verb`Generalized k-order Fibonacci and Lucas Hybrinomials.docx`  available at \url{https://authorea.com/users/355382/articles/641125-generalized-order-fibonacci-and-lucas-polynomials-and-hybrinomials}

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