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. This ladder reminds me of Kepler’s third law describing planetary motion found in his Harmonices Mundi (1619) in which the square of the periods (time for one orbit around the sun) of any two planets are proportional to the cubes of their average distance from the sun.
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
Following the P-Q modular equations used by B. Berndt (Transactions of the American Mathematical Society, 349(6), June 1997) to determine 13 radical forms of class invariants reported by Ramanujan, this Chapter discusses a new method of analysis. By using the q octic continued fraction and the q cubic solution equation (qkQ) derived in Chapter 28, all 13 radical forms are solved using results from Berndt’s two theorems. The radical forms for discriminants with prime divisors of 5 and 7 are presented without the need for modular P-Q equations.
A theorem of existence of a solution of the class invariant in radicals is presented based on an invariant of the class invariant and modulus of the octic q continued fraction.
expressing the octic q continued fraction in radical form using a q cubic solution equation
This appendix illustrates a calculation of one of Weber’s class invariants and compares the result obtained by Weber with the analysis presented in Chapter 28. The analytical method provides some extra parameters for finding a suitable irreducible polynomial that can be expressed in radical form. Consequently, the modulus of the q continued fraction is also expressed in radicals and/or nested radicals.
calculating a class invariant from ramanujan’s octic q continued fraction
This board is an extension of Chapter 20 on discriminants and modular functions. It provides a method of expressing the Rogers-Ramanujan octic q continued fraction in terms of radicals. Some interesting relations of the q continued fraction with the plastic number are shown. A q modulus equation is also derived to find new radical expressions for polynomials with various complex quadratic fields. Radical form is possible if the order of the associated polynomial with discrimination (-d) is less than five or provided the real root U of f(x) is solvable by radicals. Unexpectedly the later is true for all but four of the discriminants 1 or 3 mod 4 less than 100 using a radical extension of the irreducible coefficient field.
expressing a ramanujan q continued fraction in terms of radicals_
A look back at Chapter 23 reveals a new way to view the orthogonal Jacobi Polynomial. Through a series of identities it is possible to reduce any Jacobi polynomial P[m,a,b,x], with b> 1 and evaluated at x = 3 to a series of Jacobi polynomials with b = 1 or b = 0. The scalar vector product with the diagonal of the Chebychev T polynomial is introduced. It is shown that any Jacobi Polynomial P[m,a,b,3] is represented as vector products of an associated (m-1) row of the Pascal triangle and a shifted cycle index of the Symmetry group S(m). A combinatoric role of the Jacobi is suggested in this analysis.
The Jacobi polynomial can also be used to calculate the nth term of the Perrin sequence and the sigma orbit of prime numbers as described in Chapter 13.
The Jacobi Polynomial Revisited
The Chebyshev Orthogonal Collocation Method is used to solve the Electrode problem introduced in Chapter 25. The derivative and second derivative operators are derived from a integer sequence and a matrix transform using Chebyshev polynomials.
The Laplace Operator_2
This Chapter is a detour from the discussion of integer sequences. Instead, it describes a solution to the problem of the electrical potential in a membrane of mixed boundary conditions. We derive the spectrum of the Laplace operator in an orthonormal basis using a decomposition of the 2 dimensional operator into a set of ordinary 1st order differential equations.