Rational Functions and Association Scheme Parameters

The parameters of metric, cometric, symmetric association schemes with q 6= ±1 (the same as the parameters of the underlying orthogonal polynomials) can be given in general by evaluating a single rational function of degree (4, 4) in the complex variable q j . But in all known examples, save the simple n-gons, these reduce to polynomials of degree at most 2 in q j with q an integer. One reason this occurs is that the rational function can have singularities at points which would determine some of the parameters. This paper deals with the case in which not all of the singularities are removable, thus giving some reason why the n-gons might naturally be the only exceptions to schemes with parameters being polynomials of degree at most 2 in q j , except possibly for schemes of very small diameter.