Hypergeometric Orthogonal Polynomials Research Articles

Continued Fractions, Orthogonal Polynomials and Dirichlet Series

Using an experimental mathematics approach, new relations are obtained between Dirichlet-like series for certain periodic coefficients and the moments of certain families of orthogonal polynomials. In addition to the classical hypergeometric orthogonal polynomials, of Racah type and continuous dual Hahn type, a new similar family of orthogonal polynomials intervenes.

Continuous −1$-1$ hypergeometric orthogonal polynomials

Continuous −1$-1$ hypergeometric orthogonal polynomials

Another type of forward and backward shift relations for orthogonal polynomials in the Askey scheme

The forward and backward shift relations are basic properties of the (basic) hypergeometric orthogonal polynomials in the Askey scheme (Jacobi, Askey-Wilson, q-Racah, big q-Jacobi etc.) and they are related to the factorization of the differential or difference operators. Based on other factorizations, we obtain another type of forward and backward shift relations. Essentially, these shift relations shift only the parameters.

Toda and Laguerre–Freud equations and tau functions for hypergeometric discrete multiple orthogonal polynomials

In this paper, the authors investigate the case of discrete multiple orthogonal polynomials with two weights on the step line, which satisfy Pearson equations. The discrete multiple orthogonal polynomials in question are expressed in terms of τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions, which are double Wronskians of generalized hypergeometric series. The shifts in the spectral parameter for type II and type I multiple orthogonal polynomials are described using banded matrices. It is demonstrated that these polynomials offer solutions to multicomponent integrable extensions of the nonlinear Toda equations. Additionally, the paper characterizes extensions of the Nijhoff–Capel totally discrete Toda equations. The hypergeometric τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions are shown to provide solutions to these integrable nonlinear equations. Furthermore, the authors explore Laguerre–Freud equations, nonlinear equations for the recursion coefficients, with a particular focus on the multiple Charlier, generalized multiple Charlier, multiple Meixner II, and generalized multiple Meixner II cases.

Laguerre–Freud Equations for the Gauss Hypergeometric Discrete Orthogonal Polynomials

The Cholesky factorization of the moment matrix is considered for the Gauss hypergeometric discrete orthogonal polynomials. This family of discrete orthogonal polynomials has a weight with first moment given by ρ0=2F1a,bc+1;η. For the Gauss hypergeometric discrete orthogonal polynomials, also known as generalized Hahn of type I, Laguerre–Freud equations are found, and the differences with the equations found by Dominici and by Filipuk and Van Assche are provided.

On some hypergeometric Sobolev orthogonal polynomials with several continuous parameters

In this paper we study the following hypergeometric polynomials: $$ \mathcal{P}_n(x) = \mathcal{P}_n(x;\alpha,\beta,\delta_1, \dots,\delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+2} F_{\rho+1} (-n,n+\alpha+\beta+1,\delta_1+1, \dots,\delta_\rho+1;\alpha+1,\kappa_1+\delta_1+1, \dots,\kappa_\rho+\delta_\rho+1;x), $$ and $$ \mathcal{L}_n(x) = \mathcal{L}_n(x;\alpha,\delta_1,\dots, \delta_\rho,\kappa_1,\dots,\kappa_\rho) = $$ $$ = {}_{\rho+1} F_{\rho+1} (-n,\delta_1+1,\dots,\delta_\rho+1; \alpha+1,\kappa_1+\delta_1+1,\dots,\kappa_\rho+\delta_\rho+1;x), \qquad n\in\mathbb{Z}_+, $$ where $\alpha,\beta,\delta_1,\dots,\delta_\rho\in(-1,+\infty)$, and $\kappa_1,\dots,\kappa_\rho\in\mathbb{Z}_+$, are some parameters. The natural number $\rho$ of the continuous parameters $\delta_1,\dots,\delta_\rho$ can be chosen arbitrarily large. It is seen that the special case $\kappa_1=\dots=\kappa_\rho=0$ leads to Jacobi and Laguerre orthogonal polynomials. Of course, such polynomials and more general ones appeared in the literature earlier. Our aim here is to show that polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$ are Sobolev orthogonal polynomials on the real line with some explicit matrices of measures. The importance of the orthogonality property was our main reason to concentrate our attention on polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$. Here we shall use some our tools developed earlier. In particular, it was shown recently that Sobolev orthogonal polynomials are related by a differential equation with orthogonal systems $\mathcal{A}$ of functions acting in the direct sums of usual $L^2_\mu$ spaces of square-summable (classes of the equivalence of) functions with respect to a positive measure $\mu$. The case of a unique $L^2_\mu$ is of a special interest, since it allows to use OPRL to obtain explicit systems of Sobolev orthogonal polynomials. The main problem here is \textit{to choose a suitable linear differential operator in order to get explicit representations for Sobolev orthogonal polynomials}. The proof of the orthogonality relations is then a verification of such a choice and it goes in another direction: we start from the already known polynomials to their properties. We also study briefly such properties of the above polynomials: integral representations, differential equations and location of zeros. A system of such polynomials with a kind of the bispectrality property is constructed.

Laguerre–Freud equations for three families of hypergeometric discrete orthogonal polynomials

AbstractThe Cholesky factorization of the moment matrix is considered for discrete orthogonal polynomials of hypergeometric type. We derive the Laguerre–Freud equations when the first moments of the weights are given by the 1F2, 2F2, and 3F2 generalized hypergeometric series.

Bidiagonal factorization of tetradiagonal matrices and Darboux transformations

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. These type of matrices are oscillatory. In this paper the Lima–Loureiro hypergeometric multiple orthogonal polynomials and the Jacobi–Piñeiro multiple orthogonal polynomials are discussed at the light of this bidiagonal factorization for tetradiagonal matrices. The Darboux transformations of tetradiagonal Hessenberg matrices is studied and Christoffel formulas for the elements of the bidiagonal factorization are given, i.e., the bidiagonal factorization is given in terms of the recursion polynomials evaluated at the origin.

Quadrature formulas for Bessel polynomials

A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring’s problem in number theory and spherical designs in algebraic combinatorics. Sawa and Uchida proved the existence and the non-existence of certain rational quadrature formulas for the weight functions of certain classical orthogonal polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In this article, we extend the work of Sawa and Uchida to the weight function of the Bessel polynomials. In the proofs, we use the Riesz–Shohat theorem and Newton polygons. It is also of number theoretic interest that proofs of some results are reduced to determining the sets of rational points on elliptic curves.

Generalizations of generating functions for basic hypergeometric orthogonal polynomials

We derive generalized generating functions for basic hypergeometric orthogonal polynomials by applying connection relations with one extra free parameter to them. In particular, we generalize generating functions for the continuous \(q\)-ultraspherical/Rogers, little \(q\)-Laguerre/Wall, and \(q\)-Laguerre polynomials. Depending on what type of orthogonality these polynomials satisfy, we derive corresponding definite integrals, infinite series, bilateral infinite series, and \(q\)-integrals.

Rényi Entropies of Multidimensional Oscillator and Hydrogenic Systems with Applications to Highly Excited Rydberg States.

The various facets of the internal disorder of quantum systems can be described by means of the Rényi entropies of their single-particle probability density according to modern density functional theory and quantum information techniques. In this work, we first show the lower and upper bounds for the Rényi entropies of general and central-potential quantum systems, as well as the associated entropic uncertainty relations. Then, the Rényi entropies of multidimensional oscillator and hydrogenic-like systems are reviewed and explicitly determined for all bound stationary position and momentum states from first principles (i.e., in terms of the potential strength, the space dimensionality and the states's hyperquantum numbers). This is possible because the associated wavefunctions can be expressed by means of hypergeometric orthogonal polynomials. Emphasis is placed on the most extreme, non-trivial cases corresponding to the highly excited Rydberg states, where the Rényi entropies can be amazingly obtained in a simple, compact, and transparent form. Powerful asymptotic approaches of approximation theory have been used when the polynomial's degree or the weight-function parameter(s) of the Hermite, Laguerre, and Gegenbauer polynomials have large values. At present, these special states are being shown of increasing potential interest in quantum information and the associated quantum technologies, such as e.g., quantum key distribution, quantum computation, and quantum metrology.

Parameter andqasymptotics of Lq‐norms of hypergeometric orthogonal polynomials

AbstractThe three canonical families of the hypergeometric orthogonal polynomials in a continuous real variable (Hermite, Laguerre, and Jacobi) control the physical wavefunctions of the bound stationary states of a great number of quantum systems [Correction added after first online publication on 21 December, 2022. The sentence has been modified.]. The algebraic ‐norms of these polynomials describe many chemical, physical, and information theoretical properties of these systems, such as, for example, the kinetic and Weizsäcker energies, the position and momentum expectation values, the Rényi and Shannon entropies and the Cramér‐Rao, the Fisher‐Shannon and LMC measures of complexity. In this work, we examine review and solve theq‐asymptotics and the parameter asymptotics (i.e., when the weight function's parameter tends towards infinity) of the unweighted and weighted ‐norms for these orthogonal polynomials. This study has been motivated by the application of these algebraic norms to the energetic, entropic, and complexity‐like properties of the highly excited Rydberg and high‐dimensional pseudo‐classical states of harmonic (oscillator‐like) and Coulomb (hydrogenic) systems, and other quantum systems subject to central potentials of anharmonic type (such as, e.g., some molecular systems) [Correction added after first online publication on 21 December, 2022. Oscillatorlike has been changed to oscillator‐like.].

Utility of integral representations for basic hypergeometric functions and orthogonal polynomials

We describe the utility of integral representations for sums of basic hypergeometric functions. In particular we use these to derive an infinite sequence of transformations for symmetrizations over certain variables which the functions possess. These integral representations were studied by Bailey, Slater, Askey, Roy, Gasper and Rahman and were also used to facilitate the computation of certain outstanding problems in the theory of basic hypergeometric orthogonal polynomials in the $q$-Askey scheme. We also generalize and give consequences and transformation formulas for some fundamental integrals connected to nonterminating basic hypergeometric series and the Askey--Wilson polynomials. We express a certain integral of a ratio of infinite $q$-shifted factorials as a symmetric sum of two basic hypergeometric series with argument $q$. The result is then expressed as a $q$-integral. Examples of integral representations applied to the derivation of generating functions for Askey--Wilson are given and as well the computation of a missing generating function for the continuous dual $q$-Hahn polynomials.

Exactly solvable discrete time birth and death processes

We present 15 explicit examples of discrete time birth and death processes which are exactly solvable. They are related to hypergeometric orthogonal polynomials of the Askey scheme having discrete orthogonality measures. Namely, they are the Krawtchouk, three different kinds of q-Krawtchouk, (dual, q)-Hahn, (q)-Racah, Al-Salam–Carlitz II, q-Meixner, q-Charlier, dual big q-Jacobi, and dual big q-Laguerre polynomials. The birth and death rates are determined by using the difference equations governing the polynomials. The stationary distributions are the normalized orthogonality measures of the polynomials. The transition probabilities are neatly expressed by the normalized polynomials and the corresponding eigenvalues. This paper is simply the discrete time versions of the known solutions of the continuous time birth and death processes.

Tight relative t-designs on two shells in hypercubes, and Hahn and Hermite polynomials

Relative $t$-designs in the $n$-dimensional hypercube $\mathcal{Q}_n$ are equivalent to weighted regular $t$-wise balanced designs, which generalize combinatorial $t$-$(n,k,\lambda)$ designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean $t$-designs on two concentric spheres, in this paper we discuss tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells. We show under a mild condition that such a relative $t$-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative $t$-designs in $\mathcal{Q}_n$ supported on two shells are rare for large $t$.

Algebraic and complexity‐like properties of Jacobi polynomials: Degree and parameter asymptotics

AbstractThe Jacobi polynomials conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two‐parameter weight function on the interval . The spreading of its associated probability density (i.e., the Rakhmanov density) over the support interval has been quantified, beyond the dispersion measures (moments around the origin, variance), by the algebraic (Shannon and Rényi entropies) and the monotonic complexity‐like measures of Cramér–Rao, Fisher–Shannon, and LMC (López‐Ruiz, Mancini, and Calbet) types. These quantities, however, have been often determined in an analytically highbrow, non‐handy way; specially when the degree or the parameters are large. In this work, we determine in a simple, compact form the leading term of the entropic and complexity‐like properties of the Jacobi polynomials in the two extreme situations: (; fixed ) and (; fixed ). These two asymptotics are relevant per se and because they control the physical entropy and complexity measures of the high energy (Rydberg) and high dimensional (pseudoclassical) states of many exactly, conditional exactly, and quasi‐exactly solvable quantum‐ mechanical potentials which model numerous atomic and molecular systems.

Complexity-like properties and parameter asymptotics of -norms of Laguerre and Gegenbauer polynomials

The main monotonic statistical complexity-like measures of the Rakhmanov’s probability density associated to the hypergeometric orthogonal polynomials (HOPs) in a real continuous variable, each of them quantifying two configurational facets of spreading, are examined in this work beyond the Cramér–Rao one. The Fisher–Shannon and LMC López-Ruiz–Mancini–Calvet (LMC) complexity measures, which have two entropic components, are analytically expressed in terms of the degree and the orthogonality weight-function’s parameter(s) of the polynomials. The degree and parameter asymptotics of these two-fold spreading measures are shown for the parameter-dependent families of HOPs of Laguerre and Gegenbauer types. This is done by using the asymptotics of the Rényi and Shannon entropies, which are closely connected to the -norms of these polynomials, when the weight-function’s parameter tends toward infinity. The degree and parameter asymptotics of these Laguerre and Gegenbauer algebraic norms control the radial and angular charge and momentum distributions of numerous relevant multidimensional physical systems with a spherically-symmetric quantum-mechanical potential in the high-energy (Rydberg) and high-dimensional (quasi-classical) states, respectively. This is because the corresponding states’ wavefunctions are expressed by means of the Laguerre and Gegenbauer polynomials in both position and momentum spaces.

Entropy-Like Properties and Lq-Norms of Hypergeometric Orthogonal Polynomials: Degree Asymptotics

In this work, the spread of hypergeometric orthogonal polynomials (HOPs) along their orthogonality interval is examined by means of the main entropy-like measures of their associated Rakhmanov’s probability density—so, far beyond the standard deviation and its generalizations, the ordinary moments. The Fisher information, the Rényi and Shannon entropies, and their corresponding spreading lengths are analytically expressed in terms of the degree and the parameter(s) of the orthogonality weight function. These entropic quantities are closely related to the gradient functional (Fisher) and the Lq-norms (Rényi, Shannon) of the polynomials. In addition, the degree asymptotics for these entropy-like functionals of the three canonical families of HPOs (i.e., Hermite, Laguerre, and Jacobi polynomials) are given and briefly discussed. Finally, a number of open related issues are identified whose solutions are both physico-mathematically and computationally relevant.

A unified construction of all the hypergeometric and basic hypergeometric families of orthogonal polynomial sequences

We construct a set H of orthogonal polynomial sequences that contains all the families in the Askey scheme and the q-Askey scheme. The polynomial sequences in H are solutions of a generalized first-order difference equation which is determined by three linearly recurrent sequences of numbers. Two of these sequences are solutions of the difference equation sk+3=z(sk+2−sk+1)+sk, where z is a complex parameter, and the other sequence satisfies a related difference equation of order five.We obtain explicit expressions for the coefficients of the orthogonal polynomials and for the generalized moments with respect to a basis of Newton type of the space of polynomials. We also obtain explicit formulas for the coefficients of the three-term recurrence relation satisfied by the polynomial sequences in H.The set H contains all the 15 families in the Askey scheme of hypergeometric orthogonal polynomials [8, p. 183] and all the 29 families of basic hypergeometric orthogonal polynomial sequences in the q-Askey scheme [8, p. 413]. Each of these families is obtained by direct substitution of appropriate values for the parameters in our general formulas. The only cases that require some limits are the Hermite and continuous q-Hermite polynomials. We present the values of the parameters for some of the families.

Evaluations of certain Catalan-Hankel Pfaffians via classical skew orthogonal polynomials

This paper is to evaluate certain Catalan-Hankel Pfaffians by the theory of skew orthogonal polynomials. Due to different kinds of hypergeometric orthogonal polynomials underlying the Askey scheme, we explicitly construct the classical skew orthogonal polynomials and then give different examples of Catalan-Hankel Pfaffians with continuous and q-moment sequences.