Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called (p, q)‐Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving (p, q)‐Fibonacci polynomials.

In [Acta Math. Hungar. 66 (1995), 337–343], Bell and Daif proved that if R is a prime ring admitting a nonzero derivation such that d ( x y ) = d ( y x ) ${d(xy)=d(yx)}$ for all x , y ∈ R ${x,y\in R}$ , then R is commutative. The objective of this paper is to examine similar problems when the ring R is equipped with involution. It is shown that if a prime ring R with involution * of a characteristic different from 2 admits a nonzero derivation d such that d ( x x * ) = d ( x * x ) ${d(xx^*)=d(x^*x)}$ for all x ∈ R and S ( R ) ∩ Z ( R ) ≠ ( 0 ) ${S(R)\cap Z(R)\ne (0)}$ , then R is commutative. Moreover, some related results have also been discussed.

In this paper we define k-order Gaussian Fibonacci polynomials with boundary conditions and give the generating function, explicit formula and some identities for k-order Gaussian Fibonacci polynomials. We introduce the matrix represent and we obtain the k-order Gaussian Fibonacci Polynomials matrix. We define a new coding theory called k-order Gaussian Fibonacci Polynomials coding theory and establish the code elements for values of k. This coding/decoding method bound to the Q_k (x), R_k (x) and E_k,n (x) matrices. So, this method is different from the classical algebraic coding. Consequently, with this method, we move the coding theory onto a complex space which is a different field. Therefore, new working areas are created.

In this study, we describe the generalized k-order Fibonacci and Lucas numbers and give some important results with specific choices. The main purpose of this study is to define the generalized k-order Fibonacci hybrid and Lucas hybrid numbers and give some of their properties. With this definition, we obtain some important number sequences by making special choices. For example, we give the Horadam hybrid number for k=2 and the other series of hybrid numbers, Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal end Jacobsthal-Lucas hybrid numbers. We give the generating functions and some properties about the generalized k-order Fibonacci hybrid and Lucas hybrid numbers. Also, we identify and prove the matrix representation for the generalized k-order Fibonacci hybrid and Lucas hybrid numbers. The Qk matrix given for k-order Fibonacci numbers are defined for the generalized k-order Fibonacci hybrid and Lucas hybrid numbers and after the matrices with the generalized k-order Fibonacci hybrid and Lucas hybrid numbers are obtained with help of auxiliary matrices, important relationships and identities are established.