Related papers: Algebraic Analysis of the Hypergeometric Function …
In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type $A_1$. We introduce a notion of root basis for these root systems, and using a unique…
We consider one-point commuting difference operators of rank one. The coefficients of these operators depend on a functional parameter, shift operators being included only with positive degrees. We study these operators in the case of…
The main goal of this paper is to derive a number of identities for the generalized hypergeometric function evaluated at unity and for certain terminating multivariate hypergeometric functions from the symmetries and other properties of…
We derive one-point functions of the loop operators of Hermitian matrix-chain models at finite $N$ in terms of differential operators acting on the partition functions. The differential operators are completely determined by recursion…
We investigate the algebraic genericity of various families of continuous functions exhibiting extreme irregularity, focusing on fractal dimensions, H\"older regularity, and fractional differentiability. Our first main result shows that for…
Finite hypergeometric functions are complex valued functions on finite fields which are the analogue of the classical analytic hypergeometric functions. From the work of N.M.Katz it follows that their values are traces of Frobenius on…
A certain class of matrix-valued Borel matrix functions is introduced and it is shown that all functions of that class naturally operate on any operator T in a finite type I von Neumann algebra M in a way such that uniformly bounded…
We present a holographic formula relating functional determinants: the fermion determinant in the one-loop effective action of bulk spinors in an asymptotically locally AdS background, and the determinant of the two-point function of the…
We develop the basic theory of matrix-valued Weyl-Titchmarsh M-functions and the associated Green's matrices for whole-line and half-line self-adjoint Hamiltonian finite difference systems with separated boundary conditions.
We extend Lawvere-Pitts prop-categories (aka. hyperdoctrines) to develop a general framework for providing "algebraic" semantics for nonclassical first-order logics. This framework includes a natural notion of substitution, which allows…
Recently, we formulated the $q$-Garnier system and its variations as translations of an extended affine Weyl group of type $A^{(1)}_{2n+1}\times A^{(1)}_1\times A^{(1)}_1$. On the other hand, those systems admit particular solutions in…
We analyze the Wigner function constructed on the basis of the discrete rotation and displacement operators labeled with elements of the underlying finite field. We separately discuss the case of odd and even characteristics and analyze the…
We review properties of closed meromorphic $1$-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, the existence of separatrices, and resolution of singularities…
In this paper, we show that the generalized hypergeometric function mF_m-1 has a one parameter group of local symmetries, which is a conjugation of a flow of a rational Calogero-Mozer system. We use the symmetry to construct fermionic…
The A-hypergeometric system studied by I.M. Gelfand, M.I. Graev, A.V. Zelevinsky and the author, is defined for a set A of characters of an algebraic torus. In this paper we propose a generalization of the theory where the torus is replaced…
We consider a family of generic weighted arrangements of $n$ hyperplanes in $\C^k$ and show that the Gauss-Manin connection for the associated hypergeometric integrals, the contravariant form on the space of singular vectors, and the…
The concept of monogenic functions over real alternative $\ast$-algebras has recently been introduced to unify several classical monogenic (or regular) functions theories in hypercomplex analysis, including quaternionic, octonionic, and…
If $\fg$ is a semisimple Lie algebra, we describe the prime factors of $\mcU(\fg)$ that have enough finite dimensional modules. The proof depends on some combinatorial facts about the Weyl group which may be of independent interest. We also…
We obtain a realization of the Lie superalgebra $D(2, 1 ; \alpha)$ in differential operators on the supercircle $S^{1|2}$ and in $4\times 4$ matrices over a Weyl algebra. A contraction of $D(2, 1 ; \alpha)$ is isomorphic to the universal…
We prove that the diagonal of any finite product of algebraic functions of the form \begin{align*} {(1-x_1-\dots-x_n)^R}, \qquad R\in\mathbb{Q}, \end{align*} is a generalized hypergeometric function, and we provide an explicit description…