Chapter 29 Expressing the Octic q Continued Fraction in Radical Form using a modified Ramanujan’s Class Invariant Method

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

Chapter 28- Expressing a Rogers-Ramanujan q Continued Fraction in Radicals


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_