The addition of phasors as mentioned in the last chapter can be simplified using the law of cosines.  When two vectors in the complex plane are added the resulting vector magnitude and argument are altered.  The summation in a complex ellipse of two conjugate vectors results in a vector along the real axis.  The law of cosines is an extension of the Pythagorean theorem for triangles not containing an angle of 90 degrees.  All phasors of complex conjugate solutions can be added based on this law.  For integer sequences, the powers of these solutions are vectors which align at 0 or 180 degrees on the complex plane. 

Board 54- Law of Cosines, Generation of Sequences from Phasors

A complex octagon is represented as a group of all polynomials with negative discriminants and at least one real root.  Elliptic functions and hypergeometric functions are used to transform polynomials with positive discriminants into the complex octagon. The properties of the complex octagon for several discriminants are presented and indicate it is a precise and well-defined representation of real roots of any polynomial. Simplified methods of finding these roots using modular functions are described.

The Octagonal Geometry of Sequences

Integer sequences are part of our mathematical and physical environment.  This book has described many different classes of sequences which have been shown to be derived from various mathematical principles.  In this chapter I return to the q-octic fraction that transforms a negative integer into a real solution of the Weber functions.  The q-octic fraction is not limited to Weber functions and it is shown that real numbers between integers can be transformed into real solutions of potentially an infinite number of polynomial sequences. Expressions are derived that can generate sequences only from the norm of the q-octic fraction suggesting the Ramanujan ladder is a continuous function of negative real numbers and calculate real solutions to polynomials of integer degree.    

Board51_New Sequences from the Weber g Class Invariant

Some further relationships between modular functions and the moduli of q- octic continued fractions (QCF) are developed.  The transforms are modular- like but remove the need for complex multiplication.  They can be applied to the calculation of the g class invariants for both odd and even integer discriminants. Some integer sequences are discussed which are associated with the transform L2.

Board49 Q Transforms and g Class Invariants

In 1858 Charles Hermite and simultaneously Leopold Kronecker published “On the Solution of the General Equation of the Fifth Degree”.  Using the elliptic function and methods available to the mid- 19^th century mathematician, the approach to solving fifth degree polynomials was finally solved after centuries of unsuccessful attempts.  Having discussed symmetry of odd order equations to solve sequence number in the last Chapter, I find a similar symmetry required to solution of the quintic.  A three-step process reflecting a workable methodology of the 1800s is demonstrated with a particular general example.

Solving the Quintic using Methods Available in the 19th Century

Under certain conditions the cycle index of symmetric group sums are shown to be equivalent to limiting Jacobi Polynomials or hypergeometric equations of special form when z=1. These sums are found to be numbers from select element sequences such as the Padovan sequence.  Expanding the space of symmetric groups by changes in two parameters g and j, allows for similar calculation of numbers from other classes of sequences.  The common theme of these sequences is the coloration of objects within various symmetric groups. Sequence numbers are shown to be expansions of the cycle indices in powers of the constant coefficient of a class of polynomials of odd order. Special non-linear recurrences occur when g is or j is an even integer. 

Cycle Index of Symmetry Groups and the jacobi Polynomial