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Szmytkowski derived a certain integral with Gegenbauer polynomials. A natural generalization is to derive lookalike integrals with Jacobi polynomials. Six methods are treated to derive the first integral. The first method should be enough…

We consider differential-algebraic equations in infinite dimensional state spaces and study, under which conditions we can associate a $C_{0}$-semigroup with such equations. We determine the right space of initial values and characterise…

We generalize the property of Jacobi-orthogonality to indefinite scalar product spaces. We compare various principles and investigate relations between Osserman, Jacobi-dual, and Jacobi-orthogonal algebraic curvature tensors. We show that…

In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil $\omega_{\lambda}=\omega_2+\lambda \omega_1=u\delta'(x-y)+\f{1}{2}u_x\delta(x-y)+\lambda\delta'(x-y)$. Deformations are generated by a sequence…

We develop a spectral analysis of a class of block Jacobi operators based on the conjugate operator method of Mourre. We give several applications including scalar Jacobi operators with periodic coefficients, a class of difference operators…

We look at periodic Jacobi matrices on trees. We provide upper and lower bounds on the gap of such operators analogous to the well known gap in the spectrum of the Laplacian on the upper half-plane with hyperbolic metric. We make some…

We describe the relation between block Jacobi matrices and minimization problems for discrete time optimal control problems. Using techniques developed for the continuous case, we provide new algorithms to compute spectral invariants of…

We study the algebras of differential operators invariant with respect to the scalar slash actions of real Jacobi groups of arbitrary rank. These algebras are non-commutative and are generated by their elements of orders 2 and 3. We prove…

We extend the formulae of classical invariant theory for the Jacobian of a genus one curve of degree $n \le 4$ to curves of arbitrary degree. To do this, we associate to each genus one normal curve of degree $n$, an $n \times n$ alternating…

Properties of the Jacobi Theta3-function and its derivatives under discrete Fourier transforms are investigated, and several interesting results are obtained. The role of modulo N equivalence classes in the theory of Theta-functions is…

We consider a special class of periodic continued fractions (called alpha-fractions) and discuss the related algebraic and geometric problems. A classical description of the Jacobi variety of a hyperelliptic curve due to Jacobi naturally…

Matrix polynomials given in an orthogonal basis are considered. Following the ideas of Mackey et al. "Vector spaces of Linearizations for Matrix Polynomials" (2006), the vec- tor spaces, called M1(P), M2(P) and DM(P), of potential…

We show that that the Jacobi-identities for a W-algebra with primary fields of dimensions 3, 4 and 5 allow two different solutions. The first solution can be identified with WA_4. The second is special in the sense that, even though…

The real Jacobi group $G^J_1(\mathbb{R})$, defined as the semi-direct product of the group ${\rm SL}(2,\mathbb{R})$ with the Heisenberg group $H_1$, is embedded in a $4\times 4$ matrix realisation of the group ${\rm Sp}(2,\mathbb{R})$. The…

Denote by $J_m$ the Jacobian variety of the hyperelliptic curve defined by the affine equation $y^2=x^m+1$ over $\mathbb{Q}$, where $m \geq 3$ is a fixed positive integer. We compute several interesting arithmetic invariants of $J_m$: its…

The notion of double depth associated with quasi-Jacobi forms allows distinguishing,within the algebra of quasi-Jacobi singular forms of index zero, certain significant subalgebras (modular-type forms, elliptic-type forms, Jacobi forms). We…

We introduce two kinds of multiple little q-Jacobi polynomials by imposing orthogonality conditions with respect to r discrete little q-Jacobi measures on the exponential lattice q^k (k=0,1,2,3,...), where 0 < q < 1. We show that these…

We study the trace class perturbations of the half-line, discrete Laplacian and obtain a new bound for the perturbation determinant of the corresponding non-self-adjoint Jacobi operator. Based on this bound, we obtain the Lieb--Thirring…

The aim of this paper is to derive (by using two operators, representable by a Jacobi matrix) a family of q-orthogonal polynomials, which turn to be dual to alternative q-Charlier polynomials. A discrete orthogonality relation and a…

This paper has three main objectives: (i) To establish an isomorphism between Jacobi forms of index $D_{2n+1}$ (lattice index) and elliptic modular forms of level $2$. (ii) To provide an explicit formula for the Fourier coefficients of…