Our previous chapters have demonstrated that various functions can be used to express integer sequences or linear recurrences. The intersystem polynomials (ISP) were shown to be equivalent to Bell Polynomials for sequence numbers of polynomials of any degree. Binomial coefficients and the incomplete Beta function could be developed to calculate Perrin numbers. These equations lead to expressions of sequence numbers using the hypergeometric functions and the specialized form of the Jacobi Polynomial. Polynomials obtained from the cycle index of symmetric groups were also shown to be equivalent to these orthogonal polynomials. In this Chapter the parent sequence and its first convolution, the element sequence of even power polynomials are two sequences that are also expressed by the Jacobi and hypergeometric functions.
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.
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.
Sequence entanglement of the element sequences is described. Entanglement is possible when the three roots of the discriminant equation are real.
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.