Study of generalized Pell and Pell-Lucas sequences and polynomials

Abstract

Two main topics are covered in this thesis: the first is to establish some new identities and properties for some sequences and polynomials, that are generalized Pell numbers and Pell polynomials, and the second is to construct combinatorial interpretations for linear recurrence sequences. In the first part, we use some properties of k-Jacobsthal polynomials to establish new identities for the k-Fibonacci and k-Lucas numbers. As applications, we use these identities to prove that some diophantine equations have infinitely many solutions. Additionally, we derive some properties of divisibility related to the k-Lucas and k-Fibonacci numbers. Following that, we focus on particular types of identities, which are exactly generalizations of the identity: F_(m_1+m_2+...+m_l ),where F_nis the n-th Fibonacci number, to any linear recurrent sequence of second order. As applications, we derive many new identities for second-order recursions and new trigonometric identities, among others. To do this, we provide two different methods, the first using some nested sums and the second using some aspects of graph theory. The second part is devoted to studying some combinatorial interpretations of general (higher-order) linear recurrence sequences. We use the Riordan arrays to obtain two combinatorial interpretations: the first using Riordan arrays and the second using weighted lattice paths. As applications, we derive a new formula that expresses explicitly the general term of any linear recurrent sequence. Some double binomial sums associated with classical recurrent sequences are obtained.