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For a simple Lie algebra $mathfrak{g}$ and an irreducible faithful representation $pi$ of $mathfrak{g}$, we introduce the Schur polynomials of $(mathfrak{g},pi)$-type. We then derive the Sato-Zhou type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of $mathfrak{g}$-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of $(mathfrak{g},pi)$-type with the coefficients being the Plucker coordinates. As an application, we provide a way of computing polynomial tau functions for the DS hierarchy. For $mathfrak{g}$ of low rank, we give several examples of polynomial tau functions, and use them to detect bilinear equations for the DS hierarchy.
Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble studied in [2
We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results o
The symbolic method is used to get explicit formulae for the products or powers of Bessel functions and for the relevant integrals.
Symbolic methods of umbral nature play an important and increasing role in the theory of special functions and in related fields like combinatorics. We discuss an application of these methods to the theory of lacunary generating functions for the Lag
The generating function of the Bannai-Ito polynomials is derived using the fact that these polynomials are known to be essentially the Racah or $6j$ coefficients of the $mathfrak{osp}(1|2)$ Lie superalgebra. The derivation is carried in a realization