On shifted Fibonacci sequences and Their polynomials

Abstract

Fibonacci sequences and their polynomials have been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions and by varying the recurrence relation and maintaining the initial conditions. In this thesis, we maintain the recurrence relation and vary initial conditions which are taken as sum of Fibonacci numbers or polynomials. The main objective of this work was to generalize Fibonacci sequences and their polynomials by r-shift operation and to determine properties of these generalized sequences and their polynomials. The specific objectives were to generalize Finonacci sequences and their polynomials , to determine properties of r-shifted Fibonacci sequences and to determine properties of r-shifted Fibonacci polynomials. To achieve the first objective, we maintain recurrence relation and vary the initial conditions by r- shift operation. To achieve the second objective we mainly use Binet’s formula and generating function of r-shifted Fibonacci sequences, mathematical induction and direct proofs, and to achieve the third objective we used Binet’s formula and generating function for r-shifted Fibonacci polynomials. Among results obtained in this thesis for both r-shifted Fibonacci numbers and polynomials are explicit sum formula, sum of first n terms, sum of first n terms with even indices, sum of first n terms with odd indices, Honsberger’s identity, and generalized identity from which we get Catalan’s identity, Cassini’s identity, and d’Ocagne’s identity. The results obtained in this study add to the already existing literature in this area of research and they are also of importance to researchers in Computer Science and other fields Mathematics.