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_