Related papers: Difference equations with elliptic coefficients an…
We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that…
We solve the problem of Fourier transformation for the one-dimensional $q$-deformed Heisenberg algebra. Starting from a matrix representation of this algebra we observe that momentum and position are unbounded operators in the Hilbert…
We study the tensor product of the {\it higher spin representations} (see the definition in Sect. 2.2) of the elliptic quantum group $E_{\tau,\eta}(sl_n)$. The transfer matrices associated with the $E_{\tau,\eta}(sl_n)$-module are exactly…
Schlesinger transformations are discrete monodromy preserving symmetry transformations of a meromorphic connection which shift by integers the eigenvalues of its residues. We study Schlesinger transformations for twisted sl_N-valued…
In the late '60s, B. Dwork studied a Frobenius structure compatible with the classical hypergeometric differential equation with parameters $\left(\frac{1}{2},\frac{1}{2} ; 1 \right)$ by analyzing behavior of solutions of the differential…
To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the…
The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling…
The representation theory of the Hopf algebroid $U_{q,x}(\widehat{sl_2})=U_{q,\lambda}(\widehat{sl_2})$ is initiated and it is established that the intertwiner between the tensor products of dynamical evaluation modules is a well-poised…
Extending the method proposed in [arXiv:1109.5524], we derive QQ-relations (functional relations among Baxter Q-functions) and T-functions (eigenvalues of transfer matrices) for fusion vertex models associated with the twisted quantum…
We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a,b,h;p,q,r)…
Folding transformation of the Painlev\'e equations is an algebraic (of degree greater than 1) transformation between solutions of different equations. In 2005 Tsuda, Okamoto and Sakai classified folding transformations of differential…
We study a certain type of multiple commutation relations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$. We show that all the coefficients in the multiple commutation relations between the $L$-operator elements are given in…
The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple…
$Q$-systems and $T$-systems are systems of integrable difference equations that have recently attracted much attention, and have wide applications in representation theory and statistical mechanics. We show that certain $\tau$-functions,…
The goal of this paper is to relate the quantum category $\mathcal{O}$ (known also as the category of modules over the mixed quantum group) at an odd root of unity to the affine Hecke category. Namely, we prove equivalences of highest…
Given a (not necessarily regular) holonomic D-module defined on the product of two complex manifolds, we prove that the associated correspondence commutes (in some sense) with the De Rham functor. We apply this result to the study of the…
A universal system of difference equations associated with a hyperelliptic curve is derived constituting the discrete analogue of the Dubrovin equations arising in the theory of finite-gap integration. The parametrisation of the solutions…
We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin…
Integral identities for Macdonald polynomials play an important role in modern mathematics and mathematical physics. Especially interesting are the Cherednik-Macdonald-Mehta (CMM) identities, with profound connections to Double Affine Hecke…
We study a system of pseudodifferential equations that is elliptic in the sense of Petrovskii on a closed compact smooth manifold. We prove that the operator generated by the system is Fredholm one on a refined two-sided scale of the…