Quantum Hurwitz numbers and Macdonald polynomials (Q2836132)

The author extends the approach to the construction of parametric families of \(r\)-functions as generating functions for weighted Hurwitz numbers which are initiated by him and Guay-Paquet and were extended to other cases by him and others. The general method is developed and used to derive an infinite parametric family of \(2D\) Toda \(r\)-functions of hypergeometric type depending also on an additional pair \((q,t)\) of quantum deformation parameters entering in the definition of the scalar product. For specific choices of the parameters defining the weight generating function the author gives specialized versions of the quantum weighted Hurwitz numbers. By making other specializations involving particular values for the pair \((q,t)\) or their limits reduce the Macdonald or Schur or Hall-Littlewood or Jack polynomials.