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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 CramĂ©râ€"Rao one. The Fisherâ€"Shannon and LMC LĂłpez-Ruizâ€"Manciniâ€"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 RĂ©nyi and Shannon entropies, which are closely connected to the L q -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. [ABSTRACT FROM AUTHOR]

We calculate analytically, for finite-size matrices, joint probability densities of ratios of level spacings in ensembles of random matrices characterized by their associated confining potential.We focus on the ratios of two spacings between three consecutive real eigenvalues, as well as certain generalizations such as the overlapping ratios. The resulting formulas are further analyzed in detail in two specific cases: the β-Hermite and the β-Laguerre cases, for which we offer explicit calculations for small N. The analytical results are in excellent agreement with numerical simulations of usual random matrix ensembles, and with the level statistics of a quantum many-body lattice model and zeros of the Riemann zeta function. [ABSTRACT FROM AUTHOR]

We consider the product of M quadratic random matrices with complex elements and no further symmetry, where all matrix elements of each factor have a Gaussian distribution. This generalizes the classical Wishart-Laguerre Gaussian unitary ensemble with M = 1. In this paper, we first compute the joint probability distribution for the singular values of the product matrix when the matrix size N and the number M are fixed but arbitrary. This leads to a determinantal point process which can be realized in two different ways. First, it can be written as a one-matrix singular value model with a nonstandard Jacobian, or second, for M ⩾ 2, as a two-matrix singular value model with a set of auxiliary singular values and a weight proportional to the Meijer G-function. For both formulations, we determine all singular value correlation functions in terms of the kernels of biorthogonal polynomials which we explicitly construct. They are given in terms of the hypergeometric and Meijer G-functions, generalizing the Laguerre polynomials for M = 1. Our investigation was motivated from applications in telecommunication of multilayered scattering multiple-input and multiple-output channels.We present the ergodic mutual information for finite-N for such a channel model with M - 1 layers of scatterers as an example. [ABSTRACT FROM AUTHOR]

For 11 examples of one-dimensional quantum mechanics with shape-invariant potentials, the Darboux-Crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to Krein-Adler transformations, deleting multiple eigenstates with shifted parameters. These are based upon infinitely many polynomial Wronskian identities of classical orthogonal polynomials, i.e. the Hermite, Laguerre and Jacobi polynomials, which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape-invariant potentials. [ABSTRACT FROM AUTHOR]

Any object, described by a target function that need not be band-limited, can be sampled at any chosen set of points and then propagated without evanescent waves, using a function which is band-limited, so as to be imaged exactly (i.e. nonparaxially) at multiples of a given repetition distance. If the samples span a sub-wavelength region, the repeated images are superoscillatory. The number N of samples is equal to the repetition distance measured in wavelengths and also to the number of plane waves in the propagating field. If N>>1, the waves form a quasi-continuum, and asymptotics enables an almost-explicit description of the superoscillations. But the matrix involved is ill-conditioned (many of its eigenvalues are very small), so this method of sub-wavelength imaging would be pathologically sensitive to noise, and the depth of focus is exponentially small. [ABSTRACT FROM AUTHOR]

The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is superintegrable and its symmetry generators are obtained by the Schwinger construction using parabosonic creation/annihilation operators. The algebra generated by the constants of motion, which we term the Schwinger--Dunkl algebra, is an extension of the Lie algebra u(2) with involutions. The system admits separation of variables in both Cartesian and polar coordinates. The separated wavefunctions are respectively expressed in terms of generalized Hermite polynomials and products of Jacobi and Laguerre polynomials. Moreover, the so-called Jacobi--Dunkl polynomials appear as eigenfunctions of the symmetry operator responsible for the separation of variables in polar coordinates. The expansion coefficients between the Cartesian and polar bases (overlap coefficients) are given as linear combinations of dual -- 1 Hahn polynomials. The connection with the Clebsch--Gordan problem of the sl-1(2) algebra is explained. [ABSTRACT FROM AUTHOR]

As the third stage of the project multi-indexed orthogonal polynomials, we present, in the framework of 'discrete quantum mechanics' with pure imaginary shifts in one dimension, themulti-indexed Wilson and Askey-Wilson polynomials. They are obtained from the original Wilson and Askey-Wilson polynomials by multiple applications of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of 'virtual state solutions' of types I and II, in a similar way to the multi-indexed Laguerre, Jacobi and (q-)Racah polynomials reported earlier. [ABSTRACT FROM AUTHOR]

A method is presented that makes it possible to embed a subgroup separable superintegrable system into an infinite family of systems that are integrable and exactly-solvable. It is shown that in two dimensional Euclidean or pseudo-Euclidean spaces the method also preserves superintegrability. Two infinite families of classical and quantum superintegrable systems are obtained in twodimensional pseudo-Euclidean space whose classical trajectories and quantum eigenfunctions are investigated. In particular, the wave-functions are expressed in terms of Laguerre and generalized Bessel polynomials. [ABSTRACT FROM AUTHOR]

As the second stage of the project multi-indexed orthogonal polynomials, we present, in the framework of 'discrete quantum mechanics' with real shifts in one dimension, the multi-indexed (q-)Racah polynomials. They are obtained from the (q-)Racah polynomials by the multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of 'virtual state' vectors, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier. The virtual state vectors are the 'solutions' of the matrix Schrödinger equation with negative 'eigenvalues', except for one of the two boundary points. [ABSTRACT FROM AUTHOR]

We study a parameterized family of Toeplitz operators in the context of affine coherent states based on the Calderón reproducing formula (=resolution of unity on L2(R)) and the specific admissible wavelets (=affine coherent states in L2(R)) related to Laguerre functions. Symbols of such Calderón-Toeplitz operators as individual coordinates of the affine group (=upper half-plane with the hyperbolic geometry) are considered. In this case, a certain class of pseudo-differential operators, their properties and their operator algebras are investigated. As a result of this study, the Fredholm symbol algebras of the Calderón-Toeplitz operator algebras for these particular cases of symbols are described. [ABSTRACT FROM AUTHOR]

We show that the coefficients of the three-term recurrence relation for orthogonal polynomials with respect to a semi-classical extension of the Laguerre weight satisfy the fourth Painlev'e equation when viewed as functions of one of the parameters in the weight. We compare different approaches to derive this result, namely, the ladder operators approach, the isomonodromy deformations approach and combining the Toda system for the recurrence coefficients with a discrete equation. We also discuss a relation between the recurrence coefficients for the Freud weight and the semi-classical Laguerre weight and show how it arises from the Bäcklund transformation of the fourth Painlevé equation. [ABSTRACT FROM AUTHOR]

Matrices ... and ... of arbitrary rank N, given by simple expressions in terms of the N zeros of certain Laguerre or para-Jacobi polynomials of degree N, feature a Diophantine property. In the Laguerre case, this property states that the 2N zeros of the polynomial p2N (λ) = det [λ2+ λ... +...]are all integers; indeed we conjecture that det [λ2+ λ... + ... ] = ΠNk=1 (λ2 - k2). The results in the para-Jacobi case are somewhat analogous; they refer to the zeros of the general solution of the ODE generally characterizing Jacobi polynomials, in the special case in which this solution is a para-Jacobi polynomial featuring an additional, arbitrary parameter. [ABSTRACT FROM AUTHOR]

The first two terms in the large N asymptotic expansion of the β moment of the characteristic polynomial for the Gaussian and Laguerre β-ensembles are calculated. This is used to compute the asymptotic expansion of the spectral density in these ensembles, in the exponentially small region outside the leading support, up to terms o(1) . The leading form of the right tail of the distribution of the largest eigenvalue is given by the density in this regime. It is demonstrated that there is a scaling from this, to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge. [ABSTRACT FROM AUTHOR]

This paper provides the generalization of the work by Floreanini et al(1993 J. Phys. A: Math. Gen.26611-4) who generated bibasic hypergeometric functions from (p, q)-oscillators. We consider a six-parameter deformed oscillator algebra realized from the (p, q)-deformed boson oscillators. We build the corresponding Fock space representation in an infinite-dimensional subspace of the Hilbert space of a harmonic oscillator. We also discuss the properties of a discrete spectrum of the Hamiltonian of the deformed harmonic oscillator corresponding to this system. We then define a realization of the deformed algebra in terms of a generalized derivative and investigate the relation between this representation and generalized bibasic Laguerre functions and polynomials. [ABSTRACT FROM AUTHOR]

We present a detailed analysis of the spin models with near-neighbours interactions constructed in our previous paper (Enciso et al2005 Phys. Lett.B 605214) by a suitable generalization of the exchange operator formalism. We provide a complete description of a certain flag of finite-dimensional spaces of spin functions preserved by the Hamiltonian of each model. By explicitly diagonalizing the Hamiltonian in the latter spaces, we compute several infinite families of eigenfunctions of the above models in closed form in terms of generalized Laguerre and Jacobi polynomials. [ABSTRACT FROM AUTHOR]

In this work, we study the influence of the Aharonov–Bohm–Coulomb (ABC) system on the relativistic and non-relativistic quantum dynamics of a charged Dirac particle with position-dependent mass (PDM). In order to solve exactly our system, we use both the left-handed and right-handed projection operators. Next, we determine the Dirac spinor and the relativistic energy spectrum, where we observe that this spinor is written in terms of the generalized Laguerre polynomials and this spectrum depends explicitly on the quantum numbers n and ml, parameters Z and that characterize the ABC system, and on the parameter that characterizes the PDM. In particular, we analyse the behavior of the energies as a function of for different values of Z. Finally, we consider the non-relativistic limit and we compared our problem with others works in the literature, where we verify that our results turn out to generalize some particular cases. [ABSTRACT FROM AUTHOR]

The determination of the physical entropies (Rényi, Shannon, Tsallis) of high-dimensional quantum systems subject to a central potential requires the knowledge of the asymptotics of some power and logarithmic integral functionals of the hypergeometric orthogonal polynomials which control the wavefunctions of the stationary states. For the D-dimensional hydrogenic and oscillator-like systems, the wavefunctions of the corresponding bound states are controlled by the Laguerre () and Gegenbauer () polynomials in both position and momentum spaces, where the parameter α linearly depends on D. In this work we study the asymptotic behavior as of the associated entropy-like integral functionals of these two families of hypergeometric polynomials. [ABSTRACT FROM AUTHOR]

As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little q-Jacobi (Laguerre) polynomials in the framework of ‘discrete quantum mechanics’ with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum–Krein–Adler deletion of virtual state vectors. The virtual state vectors are the solutions of the matrix Schrödinger equation on all the lattice points having negative energies and infinite norm. This is in good contrast to the (q-)Racah systems defined on a finite lattice, in which the ‘virtual state’ vectors satisfy the matrix Schrödinger equation except for one of the two boundary points. [ABSTRACT FROM AUTHOR]

We construct new coherent states labeled by points z of the complex plane and depending on three parameters and by replacing the coefficients of the canonical coherent states by Laguerre polynomials with an order depending on γ. These coherent states are superpositions of eigenstates of the Hamiltonian with a symmetric Pöschl–Teller potential indexed by ν, which solve an ϵ-identity operator while the resolution of the identity of the states Hilbert space is acheived at the limit We obtain their wave functions in a closed form for a special case of parameters γ and ν. We also discuss their associated coherent states transform which leads to an integral representation of Hankel type for Laguerre functions. [ABSTRACT FROM AUTHOR]

We consider the interacting Bessel processes, a family of multiple-particle systems in one dimension where particles evolve as individual Bessel processes and repel each other via a log-potential. We consider two limiting regimes for this family on its two main parameters: the inverse temperature β and the Bessel index ν. We obtain the time-scaled steady-state distributions of the processes for the cases where β or ν are large but finite. In particular, for large β we show that the steady-state distribution of the system corresponds to the eigenvalue distribution of the β-Laguerre ensembles of random matrices. We also estimate the relaxation time to the steady state in both cases. We find that in the freezing regime β → ∞, the scaled final positions of the particles are locked at the square root of the zeroes of the Laguerre polynomial of parameter ν − 1/2 for any initial configuration, while in the regime ν → ∞, we prove that the scaled final positions of the particles converge to a single point. In order to obtain our results, we use the theory of Dunkl operators, in particular the intertwining operator of type B. We derive a previously unknown expression for this operator and study its behaviour in both limiting regimes. By using these limiting forms of the intertwining operator, we derive the steady-state distributions, the estimations of the relaxation times and the limiting behaviour of the processes. [ABSTRACT FROM AUTHOR]

We analyze the eigenvalue density for the Laguerre and Jacobi b-ensembles in the cases that the corresponding exponents are extensive. In particular, we obtain the asymptotic expansion up to terms o(1), in the large deviation regime outside the limiting interval of support. As found in recent studies of the large deviation density for the Gaussian b-ensemble, and Laguerre b-ensemble with fixed exponent, there is a scaling from this asymptotic expansion to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge. [ABSTRACT FROM AUTHOR]

Sequences of orthogonal polynomials on the unit circle whose Caratheodory function satisfies a Riccati-type differential equation with polynomial coefficients are studied. We deduce discrete Lax equations which lead to difference equations for the corresponding sequences of reflection parameters, and we analyze the continuous differential equations that arise when deformations through a dependence on a parameter t occur. [ABSTRACT FROM AUTHOR]

We present a new family of shape invariant potentials which could be called a 'continuous version' of the potentials corresponding to the exceptional (X) J1 Jacobi polynomials constructed recently by the present authors. In a certain limit, it reduces to a continuous family of shape invariant potentials related to the exceptional (X) L1 Laguerre polynomials. The latter was known as one example of the 'conditionally exactly solvable potentials' on a half line. [ABSTRACT FROM AUTHOR]

We consider the associated linear problem for a q-analogue of the fifth Painleve equation (q-PV). We identify a lattice of connection preserving deformations in the space of the characteristic data for the linear problem with the lattice of translational Backlund transformations for q-PV; hence, show all translational Backlund transformations possess a Lax pair. We show that the big q-Laguerre polynomials, and a suitable generalization, solve a special case of the linear problem, and hence, find solutions to q-PV in terms of determinants of Hankel matrices with entries consisting of rational or hypergeometric functions. We show that we may specialize the connection preserving deformations to rational deformations of the weight and the recurrence relations. [ABSTRACT FROM AUTHOR]

A semi-discrete Lax pair formed from the differential system and recurrence relation for semi-classical orthogonal polynomials leads to a discrete integrable equation for a specific semi-classical orthogonal polynomial weight. The main example we use is a semi-classical Laguerre weight to derive a third-order difference equation with a corresponding Lax pair. [ABSTRACT FROM AUTHOR]

In this paper we study the characteristic or generating function of a certain discontinuous linear statistic of the Laguerre unitary ensembles and show that this is a particular fifth Painlevé transcendent in the variable t, the position of the discontinuity. The proof of the ladder operators adapted to orthogonal polynomial with discontinuous weight announced sometime ago [13] is presented here, followed by the nonlinear difference equations satisfied by two auxiliary quantities and the derivation of the Painlevé equation. [ABSTRACT FROM AUTHOR]

We calculate the average of two characteristic polynomials for the real Ginibre ensemble of asymmetric random matrices, and its chiral counterpart. Considered as quadratic forms they determine a skew-symmetric kernel from which all complex eigenvalue correlations can be derived. Our results are obtained in a very simple fashion without going to an eigenvalue representation, and are completely new in the chiral case. They hold for Gaussian ensembles which are partly symmetric, with kernels given in terms of Hermite and Laguerre polynomials respectively, depending on an asymmetry parameter. This allows us to interpolate between the maximally asymmetric real Ginibre and the Gaussian orthogonal ensemble, as well as their chiral counterparts. [ABSTRACT FROM AUTHOR]

We show that skew-orthogonal functions, defined with respect to Jacobi weight wa,b(x) = (1 [?] x)a(1 + x)b, a, b > [?]1, including the limiting cases of Laguerre (wa(x) = xa e[?]x, a > [?]1) and Gaussian weight (w(x)={\rm e}^{-x^2}) , satisfy three-term recursion relation in the quaternion space. From this, we derive generalized Christoffel-Darboux (GCD) formulae for kernel functions arising in the study of the corresponding orthogonal and symplectic ensembles of random 2N × 2N matrices. Using the GCD formulae we calculate the level densities and prove that in the bulk of the spectrum, under appropriate scaling, the eigenvalue correlations are universal. We also provide evidence to show that there exists a mapping between skew-orthogonal functions arising in the study of orthogonal and symplectic ensembles of random matrices. [ABSTRACT FROM AUTHOR]

The transformation of a Laguerre series f(z) = [?][?]n=0l(a)nL(a)n(z) to a power series f(z) = [?][?]n=0gnzn is discussed. Since many nonanalytic functions can be expanded in terms of generalized Laguerre polynomials, success is not guaranteed and such a transformation can easily lead to a mathematically meaningless expansion containing power series coefficients that are infinite in magnitude. Simple sufficient conditions based on the decay rates and sign patterns of the Laguerre series coefficients l(a)n as n - [?] can be formulated which guarantee that the resulting power series represents an analytic function. The transformation produces a mathematically meaningful result if the coefficients l(a)n either decay exponentially or factorially as n - [?]. The situation is much more complicated--but also much more interesting--if the l(a)n decay only algebraically as n - [?]. If the l(a)n ultimately have the same sign, the series expansions for the power series coefficients diverge, and the corresponding function is not analytic at the origin. If the l(a)n ultimately have strictly alternating signs, the series expansions for the power series coefficients still diverge, but are summable to something finite, and the resulting power series represents an analytic function. If algebraically decaying and ultimately alternating Laguerre series coefficients l(a)n possess sufficiently simple explicit analytical expressions, the summation of the divergent series for the power series coefficients can often be accomplished with the help of analytic continuation formulae for hypergeometric series p+1Fp, but if the l(a)n have a complicated structure or if only their numerical values are available, numerical summation techniques have to be employed. It is shown that certain nonlinear sequence transformations--in particular the so-called delta transformation (Weniger 1989 Comput. Phys. Rep. 10 189-371 (equation (8.4-4)))--are able to sum the divergent series occurring in this context effectively. As a physical application of the results of this paper, the legitimacy of the rearrangement of certain one-range addition theorems for Slater-type functions (Guseinov 1980 Phy. Rev. A 22 369-71, Guseinov 2001 Int. J. Quantum Chem. 81 126-29, Guseinov 2002 Int. J. Quantum Chem. 90 114-8) is investigated. [ABSTRACT FROM AUTHOR]

In the beginning of the 1950s, Wigner introduced a fundamental deformation from the canonical quantum mechanical harmonic oscillator, which is nowadays sometimes called a Wigner quantum oscillator or a parabose oscillator. Also, in quantum mechanics the so-called Wigner distribution is considered to be the closest quantum analogue of the classical probability distribution over the phase space. In this paper, we consider which definition for such a distribution function could be used in the case of non-canonical quantum mechanics. We then explicitly compute two different expressions for this distribution function for the case of the parabose oscillator. Both expressions turn out to be multiple sums involving (generalized) Laguerre polynomials. Plots then show that the Wigner distribution function for the ground state of the parabose oscillator is similar in behaviour to the Wigner distribution function of the first excited state of the canonical quantum oscillator. [ABSTRACT FROM AUTHOR]