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

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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.