A paradox is found between derivatives of the ISPs and discrete convolution of integer sequences. ISP derivatives are convolution operators!

InterSequencePolynomials_and_Convolution_

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# Category: Uncategorized

# Chapter 36- ISPs and Convolution

# Chapter 35- Inter Sequence Polynomials and Fermat’s Last Theorem

# Chapter 34- Calculus of Integer Sequences

# Chapter 33- Geometry of the Perrin and Padovan Sequences II

# Chapter 32- The q Octic Transform

# Elliptic Functions and the Ramanujan Octave

# Chapter 30- Ramanujan Octave, Semitones, Chords and Harmonics

A paradox is found between derivatives of the ISPs and discrete convolution of integer sequences. ISP derivatives are convolution operators!

InterSequencePolynomials_and_Convolution_

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.

“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!

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.

(updated 6/15/2019)

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.

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.