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.
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 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.
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).
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?
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.