Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials

• URL: http://arxiv.org/abs/1907.08950v4
• Date: Tue, 21 May 2024 11:12:41 GMT
• Title: Discrete orthogonality relations for multi-indexed Laguerre and Jacobi polynomials
• Authors: Choon-Lin Ho, Ryu Sasaki,
• Abstract summary: We show that they also hold for multi-indexed Laguerre and Jacobis. We show that they also hold for Krein-Adlers based on the Hermite, Laguerre and Jacobis.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also hold for multi-indexed Laguerre and Jacobi polynomials, which are new orthogonal polynomials obtained by deforming these classical orthogonal polynomials. The discrete orthogonality relations could be considered as more encompassing characterisation of orthogonal polynomials than the three term recurrence relations. As the multi-indexed orthogonal polynomials start at a positive degree $\ell_{\mathcal D}\ge1$, the three term recurrence relations are broken. The extra $\ell_{\mathcal D}$ `lower degree polynomials', which are necessary for the discrete orthogonality relations, are identified. The corresponding Christoffel numbers are determined. The main results are obtained by the blow-up analysis of the second order differential operators governing the multi-indexed orthogonal polynomials around the zeros of these polynomials at a degree $\ell_{\mathcal D}+\mathcal{N}$. %changed The discrete orthogonality relations are shown to hold for another group of `new' orthogonal polynomials called Krein-Adler polynomials based on the Hermite, Laguerre and Jacobi polynomials.

Related papers

• Tensor cumulants for statistical inference on invariant distributions [49.80012009682584]
  We show that PCA becomes computationally hard at a critical value of the signal's magnitude. We define a new set of objects, which provide an explicit, near-orthogonal basis for invariants of a given degree. It also lets us analyze a new problem of distinguishing between different ensembles.
  arXiv  Detail & Related papers  (2024-04-29T14:33:24Z)
• A Direct Reduction from the Polynomial to the Adversary Method [92.54311953850168]
  We give a simple and direct reduction from the method (in the form of a dual) to the adversary method. This shows that any lower bound in the form of a dual is an adversary lower bound of a specific form.
  arXiv  Detail & Related papers  (2023-01-24T21:37:20Z)
• An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree [79.43134049617873]
  In this paper, we demonstrate an exponential separation between exact degree and approximate quantum query for a partial function. For an alphabet size, we have a constant versus separation complexity.
  arXiv  Detail & Related papers  (2023-01-22T22:08:28Z)
• Two-body Coulomb problem and $g^{(2)}$ algebra (once again about the Hydrogen atom) [77.34726150561087]
  It is shown that if the symmetry of the three-dimensional system is $(r, rho, varphi)$, the variables $(r, rho, varphi)$ allow a separation of variable $varphi$ and eigenfunctions. Thoses occur in the study of the Zeeman effect on Hydrogen atom.
  arXiv  Detail & Related papers  (2022-12-02T20:11:17Z)
• On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials [0.0]
  This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermites. It is achieved by exploiting the intrinsic relation between third-order shape-invariant Hamiltonians and the fourth Painlev'e equation.
  arXiv  Detail & Related papers  (2022-03-10T20:45:37Z)
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators [0.618778092044887]
  We introduce the general algebras characterizing a class of higher order superintegrable systems that separate in coordinates. The construction relies on underlying Heisenberg algebras and their defining higher order ladder operators.
  arXiv  Detail & Related papers  (2022-02-27T03:33:26Z)
• Polynomial decompositions with invariance and positivity inspired by tensors [1.433758865948252]
  This framework has been recently introduced for tensor decompositions, in particular for quantum many-body systems. We define invariant decompositions of structures, approximations, and undecidability to reals. Our work sheds new light on footings by putting them on an equal footing with tensors, and opens the door to extending this framework to other product structures.
  arXiv  Detail & Related papers  (2021-09-14T13:30:50Z)
• Fast Sketching of Polynomial Kernels of Polynomial Degree [61.83993156683605]
  kernel is especially important as other kernels can often be approximated by the kernel via a Taylor series expansion. Recent techniques in sketching reduce the dependence in the running time on the degree oblivious $q$ of the kernel. We give a new sketch which greatly improves upon this running time, by removing the dependence on $q$ in the leading order term.
  arXiv  Detail & Related papers  (2021-08-21T02:14:55Z)
• Sums of Separable and Quadratic Polynomials [0.3222802562733786]
  We study separable plus quadratic (SPQ)s. We show that convex SPQ optimization problems can be solved by "small" semidefinite programs.
  arXiv  Detail & Related papers  (2021-05-11T03:26:46Z)
• Positive maps and trace polynomials from the symmetric group [0.0]
  We develop a method to obtain operator inequalities and identities in several variables. We give connections to concepts in quantum information theory and invariant theory.
  arXiv  Detail & Related papers  (2020-02-28T17:43:37Z)
• SURF: A Simple, Universal, Robust, Fast Distribution Learning Algorithm [64.13217062232874]
  SURF is an algorithm for approximating distributions by piecewises. It outperforms state-of-the-art algorithms in experiments.
  arXiv  Detail & Related papers  (2020-02-22T01:03:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.