Related papers: Toda equation and special polynomials associated w…
We study a fully noncommutative generalisation of the commutative fourth Painlev\'e equation that possesses solutions in terms of an infinite Toda system over an associative unital division ring equipped by a derivation.
We introduce a new generalization of Euler's $\varphi$-function associated with a system of polynomials of several variables. We reprove by a short direct approach certain known related identities, and study some other special cases that do…
It is known that the Whittaker functions $w(q,\lambda)$ associated to the group SL(N) are eigenfunctions of the Hamiltonians of the open Toda chain, hence satisfy a set of differential equations in the Toda variables $q_i$. Using the…
We introduce generalized Kazdan-Warner equations on Riemannian manifolds associated with a linear action of a torus on a complex vector space. We show the existence and the uniqueness of the solution of the equation on any compact…
This paper begins investigation of the concept of ``generalized $\tau$-function'', defined as a generating function of all the matrix elements of a group element $g \in G$ in a given highest-weight representation of a universal enveloping…
In this paper, we study some typical arithmetic properties of Euler's totient function of polynomials over finite fields. Especially, we study polynomial analogues of some classical conjectures about Euler's totient function, such as…
We solve a functional equation connected to the algebraic characterization of generalized information functions. To prove the symmetry of the solution, we study a related system of functional equations, which involves two homographies.…
We analyze solutions of the Toda system and establish an optimal Moser-Trudinger inequality
We discuss computations of the Thom polynomials of singularity classes of maps in the basis of Schur functions. We survey the known results about the bound on the length and a rectangle containment for partitions appearing in such Schur…
In the present paper we obtain some integrable generalisations of the Toda system generated by flat connection forms taking values in higher ${\bf Z}$--grading subspaces of a simple Lie algebra, and construct their general solutions. One…
In this article, we consider the generalised two-parameter Cauchy two-matrix model and corresponding integrable lattice equation. It is shown that with parameters chosen as $1/k_i$ when $k_i\in\mathbb{Z}_{>0}$ ($i=1,\,2$), the average…
We give an explicit combinatorial formula for the Schur expansion of Macdonald polynomials indexed by partitions with second part at most two. This gives a uniform formula for both hook and two column partitions. The proof comes as a…
We define Schur categories, $\Gamma^d \mathcal C$, associated to a $\Bbbk$-linear category $\mathcal C$, over a commutative ring $\Bbbk$. The corresponding representation categories, $\mathbf{rep}\, \Gamma^d\mathcal C$, generalize…
A general Casoratian formulation is proposed for the 2D Toda lattice equation, which involves coupled eigenfunction systems. Various Casoratian type solutions are generated, through solving the resulting linear conditions and using a…
We present the q-deformed multivariate hypergeometric functions related to Schur polynomials as tau-functions of the KP and of the two-dimensional Toda lattice hierarchies. The variables of the hypergeometric functions are the higher times…
This paper studies solutions to a singular $SU(3)$ Toda system with linear source terms on a compact Riemann surface $\Sigma$ with smooth boundaries $\partial\Sigma$. We establish the existence of solutions when the parameters are not…
Toda field theories are important integrable systems. They can be regarded as constrained WZNW models, and this viewpoint helps to give their explicit general solutions, especially when a Drinfeld-Sokolov gauge is used. The main objective…
We present a family of solvable multi-matrix models associated with an arbitrary embedded graph $\Gamma$ with a single vertex. The graph with $n$ edges is equipped with $2n$ corner matrices. The partition function of each member of the…
We construct all polynomial tau-functions of the BKP, DKP and MDKP hierarches.
We take a unifying and new approach toward polynomial and trigonometric approximation in an arbitrary number of variables, resulting in a precise and general ready-to-use tool that anyone can easily apply in new situations of interest. The…