arXiv subjects

Robert Milson

Publications and source records attributed to Robert Milson.

At least 19 recordsLinked to original sources

arXiv2025

Two Useful Facts About Generating Functions

Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function. The first constructively produces "recursion relations" for the sequence from differential operators in the formal variable having the generating function as an eigenfunction. The second allows one to determine whether the sequence is orthogonal with respect to some inner product by considering the result of taking the inner product of the generating function with itself. Examples presented to demonstrate the use and value of these methods include a sequence of numbers, a family of Exceptional Hermite Polynomials, and an example illustrating the result in a non-commutative setting.

math.CA↗

arXiv2025

Exceptional Krall polynomials

In this paper we exhibit and study a novel class of exceptional Krall orthogonal polynomials of Hermite type. This means that the polynomials in question are (i) orthogonal with respect to a Hermite-type weight; (ii) are the eigenfunctions of a higher-order differential operator; (iii) the degree sequence of the polynomial family in question is missing a finite number of degrees. Regarding the second point, unlike the known class of exceptional Hermite polynomials that satisfy a second-order eigenvalue equation, the polynomials we introduce here are not eigenfunctions of any 2nd order differential operator, but are for one of 4th order. Regarding the third point, our family does not include a polynomial of degree zero and consequently satisfies a 5th order recurrence relation instead of the classical 3-term relation.

math.CA↗

arXiv2024

Classification of exceptional Jacobi polynomials

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 operators are in one-to-one correspondence with spectral diagrams, a combinatorial object that describes the number and asymptotic behaviour at the endpoints of $(-1,1)$ of all quasi-rational eigenfunctions of the operator. With a convenient indexing scheme for spectral diagrams, explicit Wronskian and integral construction formulas are given to build the exceptional operators and polynomials from the information encoded in the spectral diagram. In the fully degenerate class $\alpha,\beta\in\mathbb N_0$ there exist exceptional Jacobi operators with an arbitrary number of continuous parameters. The classification result is achieved by a careful description of all possible rational Darboux transformations that can be performed on exceptional Jacobi operators.

math.CA↗

arXiv2023

Hybrid Search method for Zermelo's navigation problem

In this paper, we present a novel algorithm called the Hybrid Search algorithm that integrates the Zermelo's Navigation Initial Value Problem with the Ferraro-Mart\'in de Diego-Almagro algorithm to find the optimal route for a vessel to reach its destination. Our algorithm is designed to work in both Euclidean and spherical spaces and utilizes a heuristic that allows the vessel to move forward while remaining within a predetermined search cone centred around the destination. This approach not only improves efficiency but also includes obstacle avoidance, making it well-suited for real-world applications. We evaluate the performance of the Hybrid Search algorithm on synthetic vector fields and real ocean currents data, demonstrating its effectiveness and performance.

cs.SE↗

arXiv2021

Exceptional Gegenbauer polynomials via isospectral deformation

We show a method to construct isospectral deformations of classical orthogonal polynomials. The construction is based on confluent Darboux transformations, and it allows to construct Sturm-Liouville problems with polynomial eigenfunctions that have an arbitrary number of continuous parameters. We propose to call these new orthogonal polynomial systems \emph{exceptional polynomials of the second kind}. We illustrate this construction by describing the class of exceptional Gegenbauer polynomials of the second kind.

math.CA↗

arXiv2021

Complete classification of rational solutions of $A_{2n}$-Painlevé systems

We provide a complete classification and an explicit representation of rational solutions to the fourth Painlevé equation PIV and its higher order generalizations known as the $A_{2n}$-Painlevé or Noumi-Yamada systems. The construction of solutions makes use of the theory of cyclic dressing chains of Schrödinger operators. Studying the local expansions of the solutions around their singularities we find that some coefficients in their Laurent expansion must vanish, which express precisely the conditions of trivial monodromy of the associated potentials. The characterization of trivial monodromy potentials with quadratic growth implies that all rational solutions can be expressed as Wronskian determinants of suitably chosen sequences of Hermite polynomials. The main classification result states that every rational solution to the $A_{2n}$-Painlevé system corresponds to a cycle of Maya diagrams, which can be indexed by an oddly coloured integer sequence. Finally, we establish the link with the standard approach to building rational solutions, based on applying Bäcklund transformations on seed solutions, by providing a representation for the symmetry group action on coloured sequences and Maya cycles.

math-ph↗

arXiv2021

Exceptional Legendre Polynomials and Confluent Darboux Transformations

Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of "exceptional" degrees. In this paper we introduce a new construction of multi-parameter exceptional Legendre polynomials by considering the isospectral deformation of the classical Legendre operator. Using confluent Darboux transformations and a technique from inverse scattering theory, we obtain a fully explicit description of the operators and polynomials in question. The main novelty of the paper is the novel construction that allows for exceptional polynomial families with an arbitrary number of real parameters.

math.CA↗

arXiv2020

Spectral Theory of Exceptional Hermite Polynomials

In this paper we revisit exceptional Hermite polynomials from the point of view of spectral theory, following the work initiated by Lance Littlejohn. Adapting a result of Deift, we provide an alternative proof of the completeness of these polynomial families. In addition, using equivalence of Hermite Wronskians we characterize the possible gap sets for the class of exceptional Hermite polynomials.

math.CA↗

arXiv2020

Constructing Coherent States for the Rational Extensions of the Harmonic Oscillator Potential

Using the formalism of Maya diagrams and ladder operators, we describe the algebra of annihilating operators for the class of rational extensions of the harmonic oscillators. This allows us to construct the corresponding coherent state in the sense of Barut and Girardello. The resulting time-dependent function is an exact solution of the time-dependent Schrodinger equation and a joint eigenfunction of the algebra of annihilators.

math-ph↗

arXiv2020

Rational solutions of Painleve systems

Although the solutions of Painlevé equations are transcendental in the sense that they cannot be expressed in terms of known elementary functions, there do exist rational solutions for specialized values of the equation parameters. A very successful approach in the study of rational solutions to Painlevé equations involves the reformulation of these scalar equations into a symmetric system of coupled, Riccati-like equations known as dressing chains. Periodic dressing chains are known to be equivalent to the $A_N$-Painlevé system, first described by Noumi and Yamada. The Noumi-Yamada system, in turn, can be linearized as using bilinear equations and $τ$-functions; the corresponding rational solutions can then be given as specializations of rational solutions of the KP hierarchy. The classification of rational solutions to Painlevé equations and systems may now be reduced to an analysis of combinatorial objects known as Maya diagrams. The upshot of this analysis is a an explicit determinental representation for rational solutions in terms of classical orthogonal polynomials. In this paper we illustrate this approach by describing Hermite-type rational solutions of Painlevé of the Noumi-Yamada system in terms of cyclic Maya diagrams. By way of example we explicitly construct Hermite-type solutions for the PIV, PV equations and the $A_4$ Painlevé system.

math-ph↗

arXiv2020

The Adelic Grassmannian and Exceptional Hermite Polynomials

It is shown that when dependence on the second flow of the KP hierarchy is added, the resulting semi-stationary wave function of certain points in George Wilson's adelic Grassmannian are generating functions of the exceptional Hermite orthogonal polynomials. This surprising correspondence between different mathematical objects that were not previously known to be so closely related is interesting in its own right, but also proves useful in two ways: it leads to new algorithms for effectively computing the associated differential and difference operators and it also answers some open questions about them.

math.CA↗

arXiv2019

Lectures on exceptional orthogonal polynomials and rational solutions to Painlevé equations

These are the lecture notes for a course on exceptional polynomials taught at the \textit{AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications} that took place in Douala (Cameroon) from October 5-12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past ten years. In addition, some new results are presented on the construction of rational solutions to Painlevé equation PIV and its higher order generalizations that belong to the $A_{2n}^{(1)}$-Painlevé hierarchy. The construction is based on dressing chains of Schrödinger operators with potentials that are rational extensions of the harmonic oscillator. Some of the material presented here (Sturm-Liouville operators, classical orthogonal polynomials, Darboux-Crum transformations, etc.) are classical and can be found in many textbooks, while some results (genus, interlacing and cyclic Maya diagrams) are new and presented for the first time in this set of lecture notes.

math-ph↗

arXiv2019

Corrigendum on the proof of completeness for exceptional Hermite polynomials

Exceptional orthogonal polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm-Liouville problem. Antonio Durán discovered a gap in the original proof of completeness for exceptional Hermite polynomials, that has propagated to analogous results for other exceptional families. In this paper we provide an alternative proof that follows essentially the same arguments, but provides a direct proof of the key lemma on which the completeness proof is based. This direct proof makes use of the theory of trivial monodromy potentials developed by Duistermaat and Grünbaum and Oblomkov.

math.CA↗

arXiv2019

Ladder operators and rational extensions

This note presents the classification of ladder operators corresponding to the class of rational extensions of the harmonic oscillator. We show that it is natural to endow the class of rational extensions and the corresponding intertwining operators with the structure of a category REXT. The combinatorial data for this interpretation is realized as a functor MD $\to$ REXT, where MD refers to the set of Maya diagrams appropriately endowed with categorical structure. Our formalism allows us to easily reproduce and extend earlier results on ladder operators.

math-ph↗

arXiv2019

Cyclic Maya diagrams and rational solutions of higher order Painlevé systems

This paper focuses on the construction of rational solutions for the $A_{2n}$ Painlevé system, also called the Noumi-Yamada system, which are considered the higher order generalizations of PIV. In this even case, we introduce a method to construct the rational solutions based on cyclic dressing chains of Schrödinger operators with potentials in the class of rational extensions of the harmonic oscillator. Each potential in the chain can be indexed by a single Maya diagram and expressed in terms of a Wronskian determinant whose entries are Hermite polynomials. We introduce the notion of cyclic Maya diagrams and we characterize them for any possible period, using the concepts of genus and interlacing. The resulting classes of solutions can be expressed in terms of special polynomials that generalize the families of generalized Hermite, generalized Okamoto and Umemura polynomials, showing that they are particular cases of a larger family.

math-ph↗

arXiv2018

Shape invariance and equivalence relations for pseudowronskians of Laguerre and Jacobi polynomials

In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be interpreted as the general transcription of shape invariance and specific discrete symmetries acting on the parameters of the isotonic oscillator and Darboux-Poschl-Teller potential.

math.CA↗

arXiv2017

Durfee rectangles and pseudo-Wronskian equivalences for Hermite polynomials

We study an equivalence class of iterated rational Darboux transformations applied on the harmonic oscillator, showing that many choices of state adding and state deleting transformations lead to the same transformed potential. As a by-product, we derive new identities between determinants whose entries are Hermite polynomials. These identities have a combinatorial interpretation in terms of Maya diagrams, partitions and Durfee rectangles, and serve to characterize the equivalence class of rational Darboux transformations. Since the determinants have different orders, we analyze the problem of finding the minimal order determinant in each equivalence class, or equivalently, the minimum number of Darboux transformations. The solution to this problem has an elegan graphical interpretation. The results are applied to provide alternative and more efficient representations for exceptional Hermite polynomials and rational solutions of the Painlevé IV equation.

math.CA↗

arXiv2017

A Bochner type classification theorem for exceptional orthogonal polynomials

It was recently conjectured that every system of exceptional orthogonal polynomials is related to classical orthogonal polynomials by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be regarded as the extension of Bochner's result for classical orthogonal polynomials to the exceptional class. As a supplementary result, we derive a canonical form for exceptional operators based on a bilinear formalism, and prove that every exceptional operator has trivial monodromy at all primary poles.

math.CA↗