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 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.
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.
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.