Hypergeometric tau functions of the universal character hierarchy

Abstract By integrating Hamiltonian theory, Schur function theory and integrable systems, we introduce a new fermionic operator A ( α , β ) which incorporates a parameter function r ( D ). Using it, we construct a class of tau functions τ ( m , n ; x , y ; α , β ) of the Universal Character (UC) hierarchy and then demonstrate that these tau functions can be expressed by hypergeometric functions. Furthermore, by choosing specific forms of the parameter function r ( D ), we show that the tau functions reduce to ordinary hypergeometric functions and their multivariate generalizations. This work not only provides a unified fermionic framework for hypergeometric tau functions but also significantly extends the connection between integrable systems and special functions.