The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formulæ for Gauss' hypergeometric functions. We define the tau-function of such a system as the generating function of the extended monodromy symplectomorphism, using an idea recently developed. This formulation allows us to determine the dependence of the tau-function on the monodromy data. Using the explicit solution of the monodromy problem, the tau-function is then explicitly written in terms of Barnes $G$-function.
In particular, if the Fuchsian singularities are placed to $0$, $1$ and $\infty$, this gives the structure constants of the asymptotical formula of Iorgov-Gamayun-Lisovyy for solutions of Painlevé VI equation.
Generalized Brézin-Gross-Witten tau-function as a hypergeometric solution of the BKP hierarchy
本文证明了广义Brézin-Gross-Witten τ-函数是具有简单权母函数的BKP族的超几何解。我们声称，它描述了一个自旋版本的严格单调赫维茨数。本文研究了一类BKP族的超几何τ-函数族，它对应于有理权母函数。特别地，构造了割-并算子，并导出了BKP Sato Grassmannian点的显式描述。这个家族的代表可以与自旋赫维茨数的有趣家族相关联，包括单调赫维茨数的自旋版本。
Fermionic representation for basic hypergeometric functions related to Schur polynomials
We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials considered by S.Milne \cite{Milne} . For $q=1$ these functions are also known as hypergeometric functions of matrix argument which are related to zonal spherical polynomials for $GL(N,C)/U(N)$ symmetric space. We show that these multivariable hypergeometric functions are tau-functions of the KP hierarchy. At the same time they are the ratios of Toda lattice tau-functions considered by Takasaki in \cite{Tinit} , \cite{T} evaluated at certain values of higher Toda lattice times. The v…
