Recent development of special functions from the viewpoint of representation theory and integrals of complex variables

我们解决了与Selberg型积分有关的一个联系问题，在共形场理论中，Selberg型积分表现为共形块。系数用q-racah多项式表示，q-racah多项式由一族正交多项式组成。另一方面，我们发现了一个一般的框架，将连接问题归结为确定相关扭圈的交数。事实上，在广义超几何函数的情况下，我们使用这个框架来解决这个问题。给出了扭圈交数形式的Jones多项式的定义，并给出了某些纽结的Jones多项式的显式表达式。

We solve a connection problem associated with an integral of Selberg type, which appears in conformal field theory as a conformal block. The coefficients are expressed by the q-Racah polynomials, which consist of a family of orthogonal polynomials. On the other hand, we find a general framework that the connection problem is reduced to determining the intersection numbers of the associated twisted cycles. Indeed, we use this framework to solve the problem in the case of the generalized hypergeometric function. Moreover, we give a definition of the Jones polynomials in terms of intersection numbers of twisted cycles and derive explicit expressions of the Jones polynomials for some knots.