“You can’t go back and change the beginning, but you can start where you are and change the ending” – C. S. Lewis
This chapter concludes the topic of Inter sequence polynomials. Many polynomial sequences can be defined by ISPs. Among these include Lucas-Fibonacci and Perrin-Padovan polynomial sequences. The derivative of the ISP is used as a solution to second order differential equations.
Higher Order InterSequencePolynomials
A paradox is found between derivatives of the ISPs and discrete convolution of integer sequences. ISP derivatives are convolution operators!
The inter sequence polynomials (ISPs) are used to show that Fermat’s last theorem is true for n = 2. Transforming to a Lucas type sequence for the ISP, it can be shown that given a particular form of two integers, the sum of their squares is equal to a square of a sum. If the same form for these integers is used for higher powers of n it is shown that the ISPs cannot be factored into the nth power of their sum.
The ISP’s are also shown to be suitable polynomials for curve fitting of sequences to evaluate non-integer values of n. The meaning of a derivative of an integer sequence is also explained.
Inter Sequence Polynomials and Fermats Last Theorem
Pursue some path, however narrow and crooked, in which you can walk with love and reverence – Thoreau
All sequences representing monic cubic polynomials are shown to be generated by a single formula based on two modified Tribonacci sequences. These representations are multi-variable polynomials in x, y, and z and increase in the number of monomial terms with n. It is shown that these polynomials are continuous and can be integrated and differentiated. These inter-sequence polynomials (ISPs) obey the fundamental theorem of calculus and are graphically shown as surface sheets. Each sheet represents a set of sequences and are connected to the fundamental sequences described by Perrin, Lucas and Narayana and elemental repeated sequences. Sheets can be individual laminae or multiple sheets which may intersect other sheets.
Calculus of Integer Sequences_
“If you don’t know where you want to go, then it doesn’t matter which path you take” – Lewis Carroll
The lacunary Legendre Polynomial discussed by Artioli and Dattoli generates the Padovan sequence. From my previous chapter on the geometry of these sequences I show how values of this Legendre polynomial are used to calculate the Perrin number for the nth term. This chalkboard then shows that the two-dimensional polynomial can be applied to other Perrin type sequences based on cubic equations. The Narayana cows sequence is analogous to the Padovan sequence in generating the nth term of sequences. Using the same geometric construction as for the Perrin sequence, a new set of sequences are described and nth terms of these sequences obtained from a general hypergeometric function.
A solution is presented for calculating the nth term from any sequence for a general cubic polynomial. An example is show of a polynomial that calculates the 5th term of any cubic equation!
Building a Perrin Sequence
Equating the octahedral form and the elliptic modular form of the j-invariant resulted in equations between different quadratic field q-octic continued fractions. These q-octic forms are transformed through radical expressions defined by the R and C transform. Interesting properties are shown for these transforms using integer and fractional arguments. The transforms are modular-like in complex multiplication and can be applied to the proper calculation of the j-invariant for various class equations from the q-octic continued fraction.
The q-Octic Transforms
The complete elliptic integral of the first kind provides the k modulus which is shown to connect these integrals and functions to the modular functions previously used for calculating the class invariant and the octic q continued fraction. The k modulus is an algebraic integer between 0 and 1. Equations are developed for odd and even class invariants and their associated octahedron. The connection to the Elliptic Theta functions and the j-invariant is also shown.
Elliptic Functions and the Ramanujan Octave
Discussion of further investigation of the connection between the octic q fraction, class invariants, tonality and the octahedron. The Ramanujan ladder is described for any class invariant of discriminant -m. A connection to the volume and edge of the octahedron is proved.
The Ramanujan Octave, Semitones, Chords and Harmonics
The calculation of the g invariant for discriminants -4m requires an adjustment to the Ramanujan octave invariant for odd discriminants. The new invariant is shown to be useful for finding solvable polynomials if the q octic continued fraction is divided by harmonics of the 12-tone scale. The 1/12 and 1/8 powers of 12 are significant in finding monic polynomials of degree 4 or less. The numbers 8 and 12 also appear in the elements of the octahedron. This platonic solid has 8 faces, 12 edges and 6 vertices. The solid may have a natural relation to the 12-tone scale only based on the number 12 but it also extends into the mathematics of modular functions. The powers 1/12 and 1/24 are found in many modular equations such as the j-invariant where is a factor used to impose independence of the invariant for elliptic curves in any coordinate system. The powers 12, 24 and 8 appear in the Weber function relations to the j-invariant. Although other variants of the Ramanujan octave have appeared in the literature, the results discussed in this paper do not have any significance beyond these mathematical observations but still serves as an intriguing connection of mathematics to architectural geometry and music harmony.
The Ramanujan Octave and Discriminants -4m
An analogy is made between the equations derived Chapter 29 with the musical equal tempered scale. I call this the Ramanujan octave based on his octic q continued fraction and its relation to class invariants. The existence theorem of solvability is demonstrated with two examples representing the product and quotient of the modulus of the octic continued fraction. These examples show a universality to finding radical forms of the class invariants for any class number. An infinite number of semitones are shown to create the Ramanujan octave!
The Ramanujan Octave and Examples of the Existence Theorem