On Pell Functions and Pell Numbers

Abstract

Number sequences have been studied by many scientists for many years because they find applications in nature and many branches of science. The most well-known number sequence among these integer sequences is the Fibonacci sequence. Later, these integer sequences were generalized and many number sequences were defined and their properties were examined. One of these is Pell number sequences. Pell numbers can be calculated in a way similar to the recurrence relation of Fibonacci numbers. The Pell numbers Pn are defined by
                       
The first few terms of Pell numbers are 1,2,5,12,29,... . In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. In addition to being used to approximate the square root of two, Pell numbers can also be used to find square triangle numbers, create integer approximations to the right isosceles triangle, and solve certain combinatorial numbering problems. In this study, we define Pell functions on real numbers R. We give the relations between Pell functions and Pell numbers. We examine Pell functions using the concept of the p − odd and p − even functions. Additionally, if p is a Pell function, then