Fibonacci and Narayana Like Sequences Theory and Applications

Abstract

With the amazing applications of Fibonacci sequence in rabbit problem and Narayana sequence in cow problem, Fibonacci and Narayana like sequences took their deep mathematical foundation in varied fields of mathematics, science, technology and social science. This motivated me to consider Fibonacci and Narayana-like sequences in cryptography, Pell s indeterminate equations and nonlinear Riccati differential equations, along with the study of certain mathematical properties of considered sequences. This thesis consists of 12 chapters. The first chapter contains the history and development of Fibonacci and Narayana numbers and some fundamental concepts. In chapter 2, we have introduced Fibonacci-like matrix sequence. We have obtained some properties including generating function, Binet s formula, summation identities, sum of cubes of n terms and derive some matrix relations of this sequence. In chapter 3, we have introduced Narayana sequence in two parameters; namely, (s, t)-Narayana sequence. In chapter 4, Narayana matrix sequence associated with Narayana sequence is introduced. In chapter 5, we have investigated periods of Narayana numbers modulo some positive integer m. In chapter 6, circulant, skew-circulant and semi-circulant matrices of Fibonacci, Lucas, Narayana, Gaussian Fibonacci, Gaussian Lucas and Gaussian Narayana numbers are introduced. Norms of these matrices and relations among these norms are obtained. In chapter 7, 8 and 9 coding/decoding algorithms are given by using circulant matrices of Gaussian Fibonacci, Gaussian Lucas and Gaussian Narayana numbers. In chapter 10, we have obtained approximate solutions of non-linear quadratic Riccati differential equations using the Galerkin method by taking Fibonacci polynomials and then Lucas polynomials as basis functions. In chapter 11, we have solved Pell s equations. Chapter 12 deals with conclusion and future scope of work. newline