Related papers: Universal relations in asymptotic formulas for ort…

A semi-infinite weighted Hankel matrix with entries defined in terms of basic hypergeometric series is explicitly diagonalized as an operator on $\ell^{2}(\mathbb{N}_{0})$. The approach uses the fact that the operator commutes with a…

We continue studying the connection between Jacobi matrices defined on a tree and multiple orthogonal polynomials (MOPs) that was discovered previously by the authors. In this paper, we consider Angelesco systems formed by two analytic…

We show that a confluent case of the big q-Jacobi polynomials P_n(x;a,b,c;q), which corresponds to a=b=-c, leads to a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<…

Jacobi's theta relations among quartic products of theta functions are generalized to those of arbitrary $n$ products. Igusa's procedure of derivation is extended to prove such general theta relations, from which we obtain general addition…

Let $\{\hat{P}_{n}(x)\}$ be an orthonormal polynomial sequence and denote by $\{w_{n}(x)\}$ the respective sequence of functions of the second kind. Suppose the Hamburger moment problem for $\{\hat{P}_{n}(x)\}$ is determinate and denote by…

We study the asymptotic behavior of the Bergman orthogonal polynomials $(p_n)_{n=0}^{\infty}$ for a class of bounded simply connected domains $D$. The class is defined by the requirement that conformal maps $\varphi$ of $D$ onto the unit…

Let $A = (a_{j,k})_{j,k=-\infty}^\infty$ be a bounded linear operator on $l^2(\mathbb{Z})$ whose diagonals $D_n(A) = (a_{j,j-n})_{j=-\infty}^\infty\in l^\infty(\mathbb{Z})$ are almost periodic sequences. For certain classes of such…

For every system $\{ p_n(z) \}_{n=0}^\infty$ of OPRL or OPUC, we construct Sobolev orthogonal polynomials $y_n(z)$, with explicit integral representations involving $p_n$. Two concrete families of Sobolev orthogonal polynomials (depending…

In this paper we introduce and discuss some classes of orthogonal polynomials in several non-commuting variables. The emphasis is on a non-commutative version of the orthogonal polynomials on the real line. We introduce recurrence equations…

There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on $\mathbb{R}$ and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for…

$C_{\lambda}$-extended oscillator algebras are realized as generalized deformed oscillator algebras. For $\lambda = 3$, the spectrum of the corresponding bosonic oscillator Hamiltonian is shown to strongly depend on the algebra parameters.…

We characterize the spectrum of one-dimensional Jacobi operators H=aS^{+}+a^{-}S^{-}+b in l^{2}(\Z) with quasi-periodic complex-valued algebro-geometric coefficients (which satisfy one (and hence infinitely many) equation(s) of the…

We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by $\alpha>-1$, $-2<\beta<-1$. We give necessary and sufficient conditions under which a conjecture by Askey, that…

We study the monotonic behaviour of the zeros of the multiple Jacobi-Angelesco orthogonal polynomials, in the diagonal case, with respect to the parameters $\alpha,\beta$ and $\gamma$. We prove that the zeros are monotonic functions of…

We introduce a canonical operator-theoretic construction associated to a finite geometric lattice, in which a simple nonassociative ``diamond product'' on the lattice basis gives rise to a family of creation operators indexed by atoms and a…

We introduce the \emph{universal algebra} of two Poisson algebras $P$ and $Q$ as a commutative algebra $A:={\mathcal P} (P, \, Q )$ satisfying a certain universal property. The universal algebra is shown to exist for any finite dimensional…

We study intertwining operator algebras introduced and constructed by Huang. In the case that the intertwining operator algebras involve intertwining operators among irreducible modules for their vertex operator subalgebras, a number of…

In this paper we commence the study of discrete harmonic analysis associated with Jacobi orthogonal polynomials of order $(\alpha,\beta)$. Particularly, we give the solution $W^{(\alpha,\beta)}_t$, $t\ge 0$, and some properties of the heat…

We present a generic operator $J$ simply defined as a linear map not increasing the degree from the vectorial space of polynomial functions into itself and we address the problem of finding the polynomial sequences that coincide with the…

We provide a full classification scheme for exceptional Jacobi operators and polynomials. The classification contains six degeneracy classes according to whether $\alpha,\beta$ or $\alpha\pm\beta$ assume integer values. Exceptional Jacobi…