On a unified approach to homogeneous second-order linear difference equations with constant coefficients and some applications

Abstract

In this article, we establish a new closed formula for the solution of homogeneous second-order linear difference equations with constant coefficients by using matrix theory. This, in turn, gives new closed formulas concerning all sequences of this type such as the Fibonacci and Lucas sequences. Next, we show the main advantage of our formula which is based on the fact that future calculations rely on the previous ones and this is true from any desired starting point. As applications, we show that Binet’s formula, in this case, is valid for negative integers as well. Finally, new summation formulas relating elements of such sequences. As a conclusion, we present a formula for the sum of squares of Chebychev polynomials of the first and second kind.

Keywords

• difference equations
• fibonacci sequence
• lucas sequence
• pell sequence
• pell-lucas sequence
• jacobsthal sequence
• jacobsthal-lucas sequence
• chebychev polynomials
• fibonacci polynomials
• lucas polynomials
• generalized binet’s formula
• 11b39
• 65q30