Shift, Symmetry, and Asymmetry in Polynomial Sequences. Title: Shift, Symmetry, and Asymmetry in Polynomial Sequences -Author

- Charles Kusniec

## Abstract

In this study, we show the existence of three types of shifts in polynomial curves in the XY-plane that will always result in integer sequences: 1. "Eureka shift", 2. Taylor shift, and 3. "Taylor shift and fit". Offset is a particular case of integer Taylor shift values. From these polynomial shift properties, we prove that there is always an absolute offset position provided by the simplest polynomial equation of all. We define mathematically what is the simplest polynomial equation of all. When we use this simplest equation of all, we can assign to any polynomial sequence an absolute value to offset í µí± = 0. In the current literature, we have three staircase functions[19] with unit steps: 1. floor, 2. ceiling, and 3. round. We show that there is no current staircase function with a unit step to express exactly the result of the offset in the polynomial integer sequences. To properly portray the offset of polynomial integer sequences, it is necessary to create a new fourth staircase function with a unit step. Just as the floor function has a complementary function ceiling, we realized that we need to create a complementary round staircase function. We call this new mathematical staircase function "round half to zero" and abbreviate it as "roundz". This new function "roundz" rounds the half value to 0 instead of 1. It can portray exactly what happens to the symmetry point when we implement offset in a polynomial sequence of integers. Then, we prove that every polynomial equation has a reference point that we call the "symmetry point" (sp). The symmetry point is always found between the elements with the smallest values (or the highest, if changing the signal) in the polynomial sequence. From the symmetry point of any polynomial sequence of integers, we can define two types of symmetry and one type of asymmetry. We name and define the sequences asymmetry and the two types of symmetries. With the definitions of symmetry, we show that we can make some similar relationships between the properties of polynomial infinite sequences and the finite sequences of divisors of integers.