Mathematical Sciences: Orthogonal Polynomials and Some of Their Applications

这个项目将涉及几个具体的主题。其中第一个涉及威尔逊多项式--这是一类由四个参数支配的一般多项式，与经典的超几何函数有关(看起来像是这些函数的初始段)。当某些参数随界增加时，多项式的渐近展开式将被研究。这项工作的动机是组合设计中的问题。第二个研究领域涉及超几何函数的商的连分式展开。这个想法可以追溯到Gauss的工作，并继续引起人们的兴趣，因为收敛的正连分式是分母多项式与之相对的正测度的Stieltjes变换。最终，人们会想要反转变换，以找到难以捉摸的度量。另一个方向是首席研究员在Askey-Wilson积分方面工作的继续。特殊函数在复定积分计算中已有了一些值得注意的应用。在阿斯基-威尔逊的例子中，发现了一个意想不到的组合评估。目前的工作将与丹尼斯·斯坦顿合作，试图充实这一发现背后的基本想法。他们将从一类特别简单的多项式开始，并寻求将这些多项式的乘积积分与在多部图的分解中找到的基本恒等式联系起来。

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.