Related papers: Laplace, Residue, and Euler integral representatio…

We give a class of examples of $A$-hypergeometric systems that display integrality of mirror maps. Specifically, these systems have solutions $F(\lambda_1,\dots,\lambda_N) = 1$ and $\log\lambda^l + G(\lambda_1,\dots,\lambda_N)$ (for certain…

We investigate the representations of the exotic conformal Galilei algebra. This is done by explicitly constructing all singular vectors within the Verma modules, and then deducing irreducibility of the associated highest weight quotient…

There is a Rota-Baxter algebra structure on the field $A=\mathbf{k}((t))$ with $ P$ being the projection map $A=\mathbf{k}[[t]]\oplus t^{-1}\mathbf{k}[t^{-1}]$ onto $ \mathbf{k}[[ t]]$. We study the representation theory and…

This review paper contains a concise introduction to highest weight representations of infinite dimensional Lie algebras, vertex operator algebras and Hilbert schemes of points, together with their physical applications to elliptic genera…

Infinite dimensional representations of the real form U_q(u_{n,1}) of the Drinfeld--Jimbo algebra U_q(gl_{n+1}) are defined. The principal series of representations of U_q(u_{n,1}) is studied. Intertwining operators for pairs of the…

This article proposes a new approach in the treatment of the Hilbert transform and some cases of the Fourier transform whose improper integrals are principal values. This approach may be useful for teaching these issues to undergraduate…

Given a Lie superalgebra \g, we introduce several variants of the representation ring, built as subrings and quotients of the ring R_{\Z_2}(\g) of virtual \g-supermodules (up to even isomorphisms). In particular, we consider the ideal…

We investigate representations of the $\ell$-th Takiff superalgebras $\widetilde{\mathfrak g}_\ell := \widetilde{\mathfrak g}\otimes \mathbb C[\theta]/(\theta^{\ell+1})$, for $\ell>0$, associated with a basic classical and a periplectic Lie…

In this paper we study hypercomplex manifolds in four dimensions. Rather than using an approach based on differential forms, we develop a dual approach using vector fields. The condition on these vector fields may then be interpreted as Lax…

Let $K$ be a field, $Q$ a quiver, and $\mathcal{A}$ the ideal of the path algebra $KQ$ that is generated by the arrows of $Q$. We present old and new results about the representation theories of the truncations $KQ/\mathcal{A}^L$, $L \in…

Generalized Darboux-Halphen (gDH) systems, which form a versatile class of three-dimensional homogeneous quadratic differential systems (HQDS's), are introduced. They generalize the Darboux-Halphen (DH) systems considered by other authors,…

We extend prior work to derive three additional M-1-dimensional integral representations--over the interval $[0,1]$ --for products of M Slater orbitals that allows their magnitudes of coordinate vector differences (square roots of…

In this paper, we introduce a novel semi-analytical method for solving a broad class of initial value problems involving differential, integro-differential, and delay equations, including those with fractional and variable-order…

In 2021, Hu and Kim defined a new type of gamma function $\widetilde{\Gamma}(x)$ from the alternating Hurwitz zeta function $\zeta_{E}(z,x)$, and obtained some of its properties. In this paper, we shall further investigate the function…

We show new integral representations for dilogarithm and trilogarithm functions on the unit interval. As a consequence, we also prove (1) new integral representations for Apery, Catalan constants, and Legendre chi functions of order 2, 3,…

We have established novel integral representations of the Riemann zeta-function and Dirichlet eta-function based on powers of trigonometric functions and digamma function, and then use these representations to find close forms of Laurent…

In this paper we discuss the path integral representations for the coordinate systems on the complex sphere S3C. The Schroedinger equation, respectively the path integral, separates in exactly 21 orthogonal coordinate systems. We enumerate…

Two integral solutions of q-difference equations of the hypergeometric type with |q|=1 are constructed by using the double sine function. One is an integral of the Barnes type and the other is of the Euler type.

We approximate the regular solutions of the incompressible Euler equation by the solution of ODEs on finite-dimensional spaces. Our approach combines Arnold's interpretation of the solution of Euler's equation for incompressible and…

In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series. Then we apply it to obtain the closed forms of all…