# Matrix representation of (d, k) - Fibonacci polynomials

## DOI:

https://doi.org/10.31926/but.mif.2024.4.66.2.11## Keywords:

k− Fibonacci polynomials, generating function, Pascal matrix, Riordan matrix## Abstract

In this study, we define the *(d, k)*− Fibonacci polynomial and examine its properties. We give the generating function, characteristic equation, and matrix representation of this polynomial. Then we get the infinite sum for the *(d, k)*− Fibonacci polynomials. We give the relationship between *(d, k)*− Fibonacci polynomial and d− Fibonacci polynomial. Also, with the help of (d, k)− Fibonacci polynomial matrix representation and the Riordan matrix, the factorization of the Pascal matrix in two different ways is given. In addition, we define the infinite *(d, k)*− Fibonacci polynomial matrix and give their inverses. The Riordan arrays linked here help us understand patterns of number concepts and prove many theorems, as well as help us make an intuitive connection for solving combinatorial problems. Among our main goals is to combine Riordan arrays with the Fibonacci number sequence, which is the most important of the number sequences, and to expand this study to the *k*-Fibonacci number sequence, which is the general form of Fibonacci number sequences. Based on the information given above, Riordan array and Pascal matrices, which have an important place in matrix theory and combinatorics studies also derived an encoding of Pascal’s triangle in matrix form, were discussed in this study and a very different generalization of the Fibonacci number sequence was studied.