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.
This chapter continues with the expansion of orthogonal polynomials with Laguerre polynomials. The Jacobi polynomial is a expanded using the associated Laguerre polynomial. The relation of the Jacobi polynomial to Delannoy numbers is the explored. I show that the asymmetric Delannoy number can be expressed as a product of Laguerre functions. A further interpretation of this product shows a relationship the asymmetric Delannoy number D~(m,n) as the product of an (n-1) dimensional Simplex with a property vector defined as an n-dimensional coloring of m+j objects. The property vector can also be described from the cycle index polynomial of a symmetry group, S(m).
A similar analysis is performed to find the Delannoy number expressed as a Jacobi polynomial. Like the asymmetric Delannoy number the Delannoy number is expressible by Jacobi polynmials and also as a dot product of an n-1 dimensional simplex with the cycle index polynomial of a symmetry group, S(n).
The Jacobi Polynomial, Laguerre Polynomial and Delannoy numbers_
This appendix updates the theory of binary sequences from the results discussed in Chapter 13. In that chapter the Perrin sequence was found to produce a period 14 binary pattern from the Sigma orbit defined in OEIS A127687. An enhanced formula for the sigma orbit is developed for use with sequences from general cubic polynomials. The results show 6 classes of binary sequences are obtained from the the ring of polynomials of degree 3. The rules for class membership are defined.
For Perrin pseudo-primes the period 14 binary sequence predicts that pseudo-primes can occur at 2, 4 and 8 mod 14. To date for numbers <10e14 only PPP(3)= 2 mod 14 and PPP(5) = 4 mod 14 have been confirmed. Can it be determined if PPP(3)*PPP(5) = 453371887665796 = 8 mod 14 is a Perrin pseudo-prime?
Appendix 13d to Perrin Pseudoprimes
The Perrin Conjugate and the Laguerre Orthogonal Polynomial
The exponential expansion of the Perrin conjugate leads to a series like the exponential generating function for the Laguerre polynomial. This orthogonal polynomial can be used to expand any polynomial in a series of Laguerre polynomials. A summation series has been developed for the classic orthogonal polynomials. Integral representations are derived using the orthogonality of the Laguerre polynomial to find monomial terms of Legendre, Hermite and Chebyshev polynomials in terms of the Gamma function. Expansions can also be easily derived for these classical polynomials using the confluent hypergeometric function. The connection of these polynomials to symmetric functions is also demonstrated.
The Perrin Conjugate and the Laguerre Orthogonal Polynomial (2)
In the previous chapter P2(x,n) was found to be a polynomial of degree 3n and divisible by a cubic polynomial G(x). In this chapter the division is defined in the finite field of the discriminant of G(x). Limits are placed on the degree 3n when the polynomial is to be completely factored in the field.
The decomposition of an N dimensional space into invariant sub-spaces is demonstrated using Groebner basis. Similar matrices are derived from the characteristic polynomials P2(x,n) and represented by symmetric geometric shapes.
Factoring P2(x, n) over a Finite Field
The divisibility of polynomials to G(x) = x^3+Bx^2+Cx-D was derived in Chapter 21. In this appendix the divisibility is extended to conjugates of G(x) and other higher degree polynomials. A general conjecture is made regarding all cubic irreducible polynomials of negative discriminant.
Appendix to Deducing Polynomial Division Algorithms_
A computerized algebra algorithm is used to find polynomials and the integer coefficients a, b and n for which the polynomial F(x) = a x2n + bxn + Dn – (Bx2+Cx+D)n is divisible by G(x) = x3 +Bx2+Cx+D. Consequently, x is any of the real or complex roots of G such that G(x) = F(x) = 0. Also, the coefficients a, b are integers derived from the integer sequence associated with the cubic polynomial G. Higher order polynomials divisible by G(x) are also derived using a Groebner basis. It is shown that new recurrence relationships can be generated from the Groebner basis.
Chapter 21— Deducing Polynomial Division Algorithms