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- 19th 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.
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.
Attached are useful formula for those interested in calculating Ramanujan class invariants and examining the relationships to the properties of the associated octahedron, the Ramanujan octaves and the Ramanujan ladder. Many of the formula are discussed in Chapters 28 to 32 in the Perrin Chalkboard.
The previous three chapters discuss the partial Bell Polynomial and its application to integer sequences and various polynomials. A new formula using the Bell Polynomial to calculate Jacobi elliptic functions is described in this paper. The infinite polynomial expansion calculates the inverse of the elliptic integrals of the first and second kind. A single formula for the elliptic sine function, sn, is sufficient to calculate all 12 elliptic functions associated with the elliptic integral of the first kind. A corresponding formula for the elliptic integral of the second kind is also described.
These functions are used in various applications such as the geometry of ellipses and in astrophysics such as calculation of relativistic elliptical orbits of planets and comets. Examples of calculated elliptical and hyperbolic orbits such as the precession of the perihelion of the planet Mercury are shown using the Bell Polynomial (sn) function. An association of the Bell Polynomial function to the q octic and the Ramanujan ladder for Weber’s class invariants is discussed
The partial Bell Polynomial expresses polynomials of any order and is convenient to calculate parent sequences and convolutions. The Perrin sequence is used as an example showing that a convolution hierarchy of sequences of various order can be algebraically equal. The condition for algebraic equality is shown and how the calculus (derivatives) of these convolutions are determined. Linear mappings of the Parent and Element sequences are shown to be group homomorphisms.
The partial Bell Polynomial is modified and shown to represent many integer sequences in the literature. The modified Bell Polynomial and its derivative are equivalent to Inter Sequence Polynomials and are applied to calculate Perrin and Padovan numbers, respectively. A detailed investigation of how these polynomials can reproduce the integer sequence is shown. The Bell Polynomial and number are important in combinatorics, partitions and mappings. This ability to describe various mappings with higher Bell Numbers provides a better understanding of the relation of integer sequences to discrete dynamic processes.
A comparison of integer linear recurrent sequences and the parent and element sequences generated from inter sequence polynomials is presented. Bell Polynomials are shown to be connected to inter sequence polynomials. Bell Polynomials can be used to calculate ISPs of any order through a simple equation.
The convolution of sequences using ISPs and their derivatives were described in a previous Chapter (see Chapter 36). The Convolution Recurrences from Inter Sequence Polynomials (CRISP) program is designed to find the coefficients of second order and third order convolutions. The order of these sequences is found to be the sum of the orders or the component sequences. By using the ISPS it was possible to find closed forms of the recurrence coefficients of the second and third derivatives of any linear second and third order parent recurrence. This program (Mathematica) is shown in the attached pdf.
Every linear integer sequence of a cubic polynomial has an associated element sequence. The element sequence is the first derivative of the parent polynomial and easily obtained from its inter sequence polynomial. Some sequences such as the Perrin sequence have an element sequence (the Padovan sequence) which is linearly proportional after rounding to its parent sequence. This chapter discusses a method for finding this associated constant of proportion for any linear integer sequence based on the discriminant. It is found that not all sequences exhibit this proportionality and an attempt is made to find condition for existence.
Initiators are a set of numbers which can generate a sequence. This appendix provides simple cubic and quadratic polynomials for calculating the initiator sequences associated with 1st and higher order maximal independent sets. Based on the constant g defined for the hypergeometric equation, sequence values can be readily calculated for odd values of g where g = 3 represents Perrin numbers. The repeated pattern of a(n) = n for small odd n is common to these sequences and suggests a pattern in nature which leads to diversity and variation of structure. An example of a DNA sequence calculated from a higher order MIS sequence is shown (see above) to produce this variation in structure.