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 [6], Bell and Daif proved that if is a prime ring admitting a nonzero derivation such that ( ) = ( ) for all , ∈ , then is commutative. The objective of this paper is to examine similar problems when the ring is equipped with involution. It is shown that if a prime ring with involution * of a characteristic di erent from 2 admits a nonzero derivation such that ( * ) = ( * ) for all ∈ and ( ) ∩ ( ) ̸ = (0), then is commutative. Moreover, some related results have also been discussed.

The Advanced Encryption Standard (AES) is the most widely used symmetric cipher today. AES has an important place in cryptology. Finite field, also known as Galois Fields, are cornerstones for understanding any cryptography. This encryption method on AES is a method that uses polynomials on Galois fields. In this paper, we generalize the AES-like cryptology on 2 × 2 matrices. We redefine the elements of k-order Fibonacci polynomials sequences using a certain irreducible polynomial in our cryptology algorithm. So, this cryptology algorithm is called AES-like cryptology on the k-order Fibonacci polynomial matrix.

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