Connection coefficients, orthogonal polynomials and the WZ-algorithms

In this paper we explore the relationship between the coefficients in the expansion of a function f(x) in orthogonal polynomials and the coefficients for the expansion of (1-x) ^m f(x), with particular attention to the case of Jacobi polynomials. Such problems arise frequently in computational chemistry. The analysis of the situation is substantially assisted by the use of two of the so-called Wilf-Zeilberger algorithms: the algorithm zeil and the algorithm hyper. We explain these algorithms and give several examples. This revised version was published online in June 2006 with corrections to the Cover Date.