Mathematical Sciences: Orthogonal Polynomials and Some of Their Applications

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Several specific topics will be addressed in this project. The first of these concerns Wilson polynomials - these are a general class of polynomials governed by four parameters and are related to classical hypergeometric functions (appearing as seems of the initial segments of these functions). Work will be done on asymptotic expansions of polynomials as certain of the parameters increase withound bound. The work is motivated by problems in combinatorial design. A second area of research concerns continued fraction expansions of quotients of hypergeometric functions. This idea can be traced back to work of Gauss and continues to be of interest because convergent positive continued fractions are Stieltjes transforms of the positive measure against which the denominator polynomials are otrhogonal. Ultimately one will want to invert the transform to find the elusive measure. Another direction represents a continuation of the principal investigator's work on Askey-Wilson integrals. Some remarkable applications of special functions have been made to the evaluation of complex definite integrals. The Askey-Wilson example was one in which an unexpected combinatorial evaluation was discovered. The current work, in collaboration with Dennis Stanton, will seek to flesh out the fundamental ideas behind this discovery. They will begin with a particularly simple class of polynomials and seek to connect the integral of products of these polynomials with basic identities found in decompositions of multipartite graphs.