arXiv subjects
Publications and source records attributed to J. Arvesú.
We study a family of type II multiple orthogonal polynomials. We consider orthogonality conditions with respect to a vector measure, in which each component is a q-analogue of the binomial distribution. The lowering and raising operators as well as the Rodrigues formula for these polynomials are obtained. The difference equation of order r+1 is studied. The connection via limit relation between four types of Kravchuk polynomials is discussed.
We investigate interlacing properties of zeros of Laguerre polynomials $ L_{n}^{(α)}(x)$ and $ L_{n+1}^{(α+k)}(x),$ $ α> -1, $ where $ n \in \mathbb{N}$ and $ k \in {\{ 1,2 }\}$. We prove that, in general, the zeros of these polynomials interlace partially and not fully. The sharp $t-$interval within which the zeros of two equal degree Laguerre polynomials $ L_n^{(α)}(x)$ and $ L_n^{(α+t)}(x)$ are interlacing for every $n \in \mathbb{N}$ and each $ α> -1$ is $ 0 < t \leq 2,$ \cite{DrMu2}, and the sharp $t-$interval within which the zeros of two consecutive degree Laguerre polynomials $ L_n^{(α)}(x)$ and $ L_{n-1}^{(α+t)}(x)$ are interlacing for every $n \in \mathbb{N}$ and each $ α> -1$ is $ 0 \leq t \leq 2,$ \cite{DrMu1}. We derive conditions on $n \in \mathbb{N}$ and $α,$ $ α> -1$ that determine the partial or full interlacing of the zeros of $ L_n^{(α)}(x)$ and the zeros of $ L_n^{(α+ 2 + k)}(x),$ $ k \in {\{ 1,2 }\}$. We also prove that partial interlacing holds between the zeros of $ L_n^{(α)}(x)$ and $ L_{n-1}^{(α+ 2 +k )}(x)$ when $ k \in {\{ 1,2 }\},$ $n \in \mathbb{N}$ and $ α> -1$. Numerical illustrations of interlacing and its breakdown are provided.
Suppose $\{P_{n}^{(α, β)}(x)\}_{n=0}^\infty $ is a sequence of Jacobi polynomials with $ α, β>-1.$ We discuss special cases of a question raised by Alan Sokal at OPSFA in 2019, namely, whether the zeros of $ P_{n}^{(α,β)}(x)$ and $ P_{n+k}^{(α+ t, β+ s )}(x)$ are interlacing if $s,t >0$ and $ k \in \mathbb{N}.$ We consider two cases of this question for Jacobi polynomials of consecutive degree and prove that the zeros of $ P_{n}^{(α,β)}(x)$ and $ P_{n+1}^{(α, β+ 1 )}(x),$ $ α> -1, β> 0, $ $ n \in \mathbb{N},$ are partially, but in general not fully, interlacing depending on the values of $α, β$ and $n.$ A similar result holds for the extent to which interlacing holds between the zeros of $ P_{n}^{(α,β)}(x)$ and $ P_{n+1}^{(α+ 1, β+ 1 )}(x),$ $ α>-1, β> -1.$ It is known that the zeros of the equal degree Jacobi polynomials $ P_{n}^{(α,β)}(x)$ and $ P_{n}^{(α- t, β+ s )}(x)$ are interlacing for $ α-t > -1, β> -1, $ $0 \leq t,s \leq 2.$ We prove that partial, but in general not full, interlacing of zeros holds between the zeros of $ P_{n}^{(α,β)}(x)$ and $ P_{n}^{(α+ 1, β+ 1 )}(x),$ when $ α> -1, β> -1.$ We provide numerical examples that confirm that the results we prove cannot be strengthened in general. The symmetric case $α= β= λ-1/2$ of the Jacobi polynomials is also considered. We prove that the zeros of the ultraspherical polynomials $ C_{n}^{(λ)}(x)$ and $ C_{n + 1}^{(λ+1)}(x),$ $ λ> -1/2$ are partially, but in general not fully, interlacing. The interlacing of the zeros of the equal degree ultraspherical polynomials $ C_{n}^{(λ)}(x)$ and $ C_{n}^{(λ+3)}(x),$ $ λ> -1/2,$ is also discussed.
We study a new family of q-Meixner multiple orthogonal polynomials of the first kind. The discrete orthogonality conditions are considered over a non-uniform lattice with respect to different q-analogues of Pascal distributions. We address some algebraic properties, namely raising and lowering operators as well as Rodrigues-type. Based on the explicit expressions for the raising and lowering operator a high-order linear q-difference equation with polynomial coefficients for the q-Meixner multiple orthogonal polynomials of the first kind is obtained. Finally, we obtain the nearest neighbor recurrence relation based on a purely algebraic approach.
We study the asymptotic behavior of Multiple Meixner polynomials of first and second kind, respectively (see J. Arvesú et al. J. Comput. Appl. Math., 153, (2003)). We use an algebraic function formulation for the solution of the equilibrium problem with constrain to describe their zero distribution. Then analyzing the limiting behavior of the coefficients of the recurrence relations for Multiple Meixner polynomials we obtain the main term of their asymptotics.
In this paper we give a characterization of some classical q-orthogonal polynomials in terms of a difference property of the associated Stieltjes function, i.e this function solves a first order non-homogeneous q-difference equation. The solutions of the aforementioned q-difference equation (given in terms of hypergeometric series) for some canonical cases, namely, q-Charlier, q-Kravchuk, q-Meixner and q-Hahn are worked out.
New (infinitely many) rational approximants to ζ(3) proving its irrationality are given. The recurrence relations for the numerator and denominator of these approximants as well as their continued fraction expansions are obtained. A comparison of our approximants with Apéry's approximants to ζ(3) is shown.