Tag Archives: Delannoy number

Chapter 27 – The Jacobi Polynomial Revisited

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 Revisited_