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We demonstrate that certain classes of Schlömilch-like infinite series and series that include generalized hypergeometric functions can be calculated in closed form starting from a simple quantum model of a particle trapped inside an infinite potential well and using principles of quantum mechanics. We provide a general framework based on the Hilbert space eigenproblem that can be applied to different exactly solvable quantum models. Obtaining series from normalization conditions in well-defined quantum problems secures their convergence. [ABSTRACT FROM AUTHOR]

We generalise well-known integrals of Ingham-Siegel and Fisher-Hartwig type over the unitary group U(N) with respect to Haar measure, for finite N and including fixed external matrices. When depending only on the eigenvalues of the unitary matrix, such integrals can be related to a Toeplitz determinant with jump singularities. After introducing fixed deterministic matrices as external sources, the integrals can no longer be solved using Andréiéf's integration formula. Resorting to the character expansion as put forward by Balantekin, we derive explicit determinantal formulae containing Kummer's confluent and Gauß' hypergeometric function. They depend only on the eigenvalues of the deterministic matrices and are analytic in the parameters of the jump singularities. Furthermore, unitary two-matrix integrals of the same type are proposed and solved in the same manner. When making part of the deterministic matrices random and integrating over them, we obtain similar formulae in terms of Pfaffian determinants. This is reminiscent to a unitary group integral found recently by Kanazawa and Kieburg [J. Phys. A: Math. Theor. 51(34), 345202 (2018)]. [ABSTRACT FROM AUTHOR]

We discuss the monic polynomials of degree n orthogonal with respect to the perturbed Gaussian weight w (z , t) = | z | α ( z 2 + t) λ e − z 2 , z ∈ R , t > 0 , α > − 1 , λ > 0 , which arises from a symmetrization of a semi-classical Laguerre weight w Lag (z , t) = z γ (z + t) ρ e − z , z ∈ R + , t > 0 , γ > − 1 , ρ > 0. The weight wLag(z) has been widely investigated in multiple-input multi-output antenna wireless communication systems in information theory. Based on the ladder operator method, two auxiliary quantities, Rn(t) and rn(t), which are related to the three-term recurrence coefficients βn(t), are defined, and we show that they satisfy coupled Riccati equations. This turns to be a particular Painlevé V (PV, for short), i.e., P V λ 2 2 , − (1 − (− 1) n α) 2 8 , − 2 n + α + 2 λ + 1 2 , − 1 2 . We also consider the quantity σ n (t) ≔ 2 t d d t ln D n (t) , which is allied to the logarithmic derivative of the Hankel determinant Dn(t). The difference and differential equations satisfied by σn(t), as well as an alternative integral representation of Dn(t), are obtained. The asymptotics of the Hankel determinant under a suitable double scaling, i.e., n → ∞ and t → 0 such that s ≔ 4nt is fixed, are established. Finally, by using the second order difference equation satisfied by the recurrence coefficients, we obtain the large n full asymptotic expansions of βn(t) with the aid of Dyson's Coulomb fluid approach. By employing these results, the second differential equations satisfied by the orthogonal polynomials will be reduced to a confluent Heun equation. [ABSTRACT FROM AUTHOR]

We use the tridiagonal representation approach to solve the radial Schrödinger equation for the continuum scattering states of the Coulomb problem in a complete basis set of discrete Bessel functions. Consequently, we obtain a new representation of the confluent hypergeometric function as an infinite sum of Bessel functions, which is numerically very stable and more rapidly convergent than another well-known formula. [ABSTRACT FROM AUTHOR]

By putting two harmonic oscillator potentials x2 side by side with a separation 2d, two exactly solvable piecewise analytic quantum systems with a free parameter d > 0 are obtained. Due to the mirror symmetry, their eigenvalues {E} for the even and odd parity sectors are determined exactly as the zeros of certain combinations of the confluent hypergeometric function F 1 1 of d and E, which are common to VD and VS but in two different branches. The eigenfunctions are the piecewise square integrable combinations of F 1 1 , the so-called U functions. By comparing the eigenvalues and eigenfunctions for various values of the separation d, vivid pictures unfold showing the tunneling effects between the two wells. [ABSTRACT FROM AUTHOR]

By putting two harmonic oscillator potentials x2 side by side with a separation 2d, two exactly solvable piecewise analytic quantum systems with a free parameter d > 0 are obtained. Due to the mirror symmetry, their eigenvalues {E} for the even and odd parity sectors are determined exactly as the zeros of certain combinations of the confluent hypergeometric function F 1 1 of d and E, which are common to VD and VS but in two different branches. The eigenfunctions are the piecewise square integrable combinations of F 1 1 , the so-called U functions. By comparing the eigenvalues and eigenfunctions for various values of the separation d, vivid pictures unfold showing the tunneling effects between the two wells. [ABSTRACT FROM AUTHOR]

In this paper, we study the Dunkl oscillator model in a generalization of superintegrable Euclidean Hamiltonian systems to the two-dimensional curved ones with a m:n frequency ratio. This defined model of the two-dimensional curved systems depends on a curvature/deformation parameter of the underlying space involving reflection operators. The curved Hamiltonian H κ admits the separation of variables in both geodesic parallel and polar coordinates, which generalizes the Cartesian coordinates of the plane. Similar to the behavior of the Euclidean case, which is the κ → 0 limit case of the curved space, the superintegrability of a curved Dunkl oscillator is naturally understood from the factorization approach viewpoint in that setting. Therefore, their associated sets of polynomial constants of motion (symmetries) as well as algebraic relations are obtained for each of them separately. The energy spectrum of the Hamiltonian H κ and the separated eigenfunctions are algebraically given in terms of hypergeometric functions and in the special limit case of null curvature occur in the Laguerre and Jacobi polynomials. Finally, the overlap coefficients between the two bases of the geodesic parallel and polar coordinates are given by hypergeometric polynomials. [ABSTRACT FROM AUTHOR]

Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite-part of divergent integrals [E. A. Galapon, Proc. R. Soc., A 473, 20160567 (2017)]. Current application of the method involves exact and asymptotic evaluation of the generalized Stieltjes transform ∫ 0 a f (x) / (ω + x) ρ d x under the assumption that the extension of f(x) in the complex plane is entire. In this paper, the method is elaborated further and extended to accommodate the presence of competing singularities of the complex extension of f(x). Finite-part integration is then applied to derive consequences of known Stieltjes integral representations of the Gauss function and the generalized hypergeometric function that involve Stieltjes transforms of functions with complex extensions having singularities in the complex plane. Transformation equations for the Gauss function are obtained from which known transformation equations are shown to follow. In addition, building on the results for the Gauss function, transformation equations involving the generalized hypergeometric function 3F2 are derived. [ABSTRACT FROM AUTHOR]

In this paper, we determine solutions for the Lévy-Leblond operator or a parabolic Dirac operator in terms of hypergeometric functions and spherical harmonics. We subsequently generalize our approach to a wider class of Dirac operators depending on 4 parameters. [ABSTRACT FROM AUTHOR]

We review the definition of hypergeometric coherent states, discussing some representative examples. Then, we study mathematical and statistical properties of hypergeometric Schrödinger cat states, defined as orthonormalized eigenstates of kth powers of nonlinear f-oscillator annihilation operators, with f of the hypergeometric type. These "k-hypercats" can be written as an equally weighted superposition of hypergeometric coherent states ∣zl⟩, l = 0, 1, ..., k − 1, with zl = ze2πil/k a kth root of zk, and they interpolate between number and coherent states. This fact motivates a continuous circle representation for high k. We also extend our study to truncated hypergeometric functions (finite dimensional Hilbert spaces), and a discrete exact circle representation is provided. We also show how to generate k-hypercats by amplitude dispersion in a Kerr medium and analyze their generalized Husimi Q-function in the super- and sub-Poissonian cases at different fractions of the revival time. [ABSTRACT FROM AUTHOR]

The Krein-like r-functionals of the hypergeometric orthogonal polynomials {pn(x)}, with the kernel of the form x s [ ω (x) ] β p m 1 (x) ... p m r (x) being ω(x) the weight function on the interval Δ ∈ R , are determined by means of the Srivastava linearization method. The particular 2-functionals, which are particularly relevant in quantum physics, are explicitly given in terms of the degrees and the characteristic parameters of the polynomials. They include the well-known power moments and the novel Krein-like moments. Moreover, various related types of exponential and logarithmic functionals are also investigated. [ABSTRACT FROM AUTHOR]

We prove a Fredholm determinant and short-distance series representation of the Painlevé V tau function τ t associated with generic monodromy data. Using a relation of τ t to two different types of irregular c = 1 Virasoro conformal blocks and the confluence from Painlevé VI equation, connection formulas between the parameters of asymptotic expansions at 0 and i∞ are conjectured. Explicit evaluations of the connection constants relating the tau function asymptotics as t → 0, +∞, i∞ are obtained. We also show that irregular conformal blocks of rank 1, for arbitrary central charge, are obtained as confluent limits of the regular conformal blocks. [ABSTRACT FROM AUTHOR]

Multiparametric families of hypergeometric τ-functions of KP or Toda type serve as generating functions for weighted Hurwitz numbers, providing weighted enumerations of branched covers of the Riemann sphere. A graphical interpretation of the weighting is given in terms of constellations mapped onto the covering surface. The theory is placed within the framework of topological recursion, with the Baker function at t = 0 shown to satisfy the quantum spectral curve equation, whose classical limit is rational. A basis for the space of formal power series in the spectral variable is generated that is adapted to the Grassmannian element associated with the τ-function. Multicurrent correlators are defined in terms of the τ-function and shown to provide an alternative generating function for weighted Hurwitz numbers. Fermionic vacuum state expectation value representations are provided for the adapted bases, pair correlators, and multicurrent correlators. Choosing the weight generating function as a polynomial and restricting the number of nonzero “second” KP flow parameters in the Toda τ-function to be finite implies a finite rank covariant derivative equation with rational coefficients satisfied by a finite “window” of adapted basis elements. The pair correlator is shown to provide a Christoffel-Darboux type finite rank integrable kernel, and the WKB series coefficients of the associated adjoint system are computed recursively, leading to topological recursion relations for the generators of the weighted Hurwitz numbers. [ABSTRACT FROM AUTHOR]

We calculate 3-loop master integrals for heavy quark correlators and the 3-loop quantum chromodynamics corrections to the ρ-parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-N space either. The solution of the homogeneous equations is possible in terms of 2F1 Gauß hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using q-product and series representations implied by Jacobi’s ϑi functions and Dedekind’s η-function. The corresponding representations can be traced back to polynomials out of Lambert–Eisenstein series, having representations also as elliptic polylogarithms, a q-factorial 1/ηk(τ), logarithms, and polylogarithms of q and their q-integrals. Due to the specific form of the physical variable x(q) for different processes, different representations do usually appear. Numerical results are also presented. [ABSTRACT FROM AUTHOR]

The derivatives of eight Horn hypergeometric functions [four Appell F1, F2, F3, and F4, and four (degenerate) confluent Φ1, Φ2, Ψ1, and Ξ1] with respect to their parameters are studied. The first derivatives are expressed, systematically, as triple infinite summations or, alternatively, as single summations of two-variable Kampé de Fériet functions. Taking advantage of previously established expressions for the derivative of the confluent or Gaussian hypergeometric functions, the generalization to the nth derivative of Horn's functions with respect to their parameters is rather straightforward in most cases; the results are expressed in terms of n+2 infinite summations. Following a similar procedure, mixed derivatives are also treated. An illustration of the usefulness of the derivatives of F1, with respect to the first and third parameters, is given with the study of autoionization of atoms occurring as part of a post-collisional process. Their evaluation setting the Coulomb charge to zero provides the coefficients of a Born-like expansion of the interaction. [ABSTRACT FROM AUTHOR]

It is shown that the tridiagonalization of the hypergeometric operator L yields the generic Heun operator M. The algebra generated by the operators L,M and Z = [L,M] is quadratic and a one-parameter generalization of the Racah algebra. The Racah-Heun orthogonal polynomials are introduced as overlap coefficients between the eigenfunctions of the operators L and M. An interpretation in terms of the Racah problem for su(1,1) and separation of variables in a superintegrable system are discussed. [ABSTRACT FROM AUTHOR]

Parametric families in the center Z(C[Sn]) of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda τ-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sphere. Combinatorially, they give quantum weighted sums over paths in the Cayley graph of Sn generated by transpositions. Dual pairs of bases for the algebra of symmetric functions with respect to the scalar product in which the Macdonald polynomials are orthogonal provide both the geometrical and combinatorial significance of these quantum weighted enumerative invariants. [ABSTRACT FROM AUTHOR]

This article considers Whittaker's confluent hypergeometric function Wκ,μ where κ is real and μ is real or purely imaginary. Then ϕ(x) = x−μ−1/2Wκ,μ(x) arises as the scattering function of a continuous time linear system with state space L²(1/2, ∞) and input and output spaces C. The Hankel operator Γϕ on L²(0, ∞) is expressed as a matrix with respect to the Laguerre basis and gives the Hankel matrix of moments of a Jacobi weight w0(x) = xb(1 − x)a. The operation of translating ϕ is equivalent to deforming w0 to give wt(x) = e−t/xxb(1 − x)a. The determinant of the Hankel matrix of moments of wε satisfies the σ form of Painlevé's transcendental differential equation PV. It is shown that Γϕ gives rise to the Whittaker kernel from random matrix theory, as studied by Borodin and Olshanski [Commun. Math. Phys. 211, 335-358 (2000)]. Whittaker kernels are closely related to systems of orthogonal polynomials for a Pollaczek-Jacobi type weight lying outside the usual Szegö class. [ABSTRACT FROM AUTHOR]

It is shown that the application of the Nikiforov-Uvarov method by Ikhdair for solving the Dirac equation with the radial Rosen-Morse potential plus the spin-orbit centrifugal term is inadequate because the required conditions are not satisfied. The energy spectra given is incorrect and the wave functions are not physically acceptable. We clarify the problem and prove that the spinor wave functions are expressed in terms of the generalized hypergeometric functions 2F1(a, b, c; z). The energy eigenvalues for the bound states are given by the solution of a transcendental equation involving the hypergeometric function. [ABSTRACT FROM AUTHOR]

We define the generalized basic hypergeometric polynomial of degree N in terms of the generalized basic hypergeometric function, by choosing one of its parameters to allow the termination of the series after a finite number of summands. In this paper, we obtain a set of nonlinear algebraic equations satisfied by the N zeros of the polynomial. Moreover, we obtain an N × N matrix M defined in terms of the zeros of the polynomial, which, in turn, depend on the parameters of the polynomial. The eigenvalues of this remarkable matrix M are given by neat expressions that depend only on some of the parameters of the polynomial; that is, the matrix M is isospectral. Moreover, in case the parameters that appear in the expressions for the eigenvalues of M are rational, the matrix M has rational eigenvalues, a Diophantine property. [ABSTRACT FROM AUTHOR]

In thiswork, one provides a justification of the condition that is usually imposed on the parameters of the hypergeometric equation, related to the solutions of the stationary Schrödinger equation for the harmonic oscillator in two-dimensional constant curvature spaces, in order to determine the solutions which are square-integrable. One proves that in case of negative curvature, it is a necessary condition of square integrability and in case of positive curvature, a necessary condition of regularity. The proof is based on the analytic continuation formulas for the hypergeometric function. It is observed also that the same is true in case of a slightly more general potential than the one for harmonic oscillator. [ABSTRACT FROM AUTHOR]

We construct coherent states through special superpositions of eigenstates of the relativistic isotonic oscillator. In each superposition, the coefficients are chosen to be L²- eigenfunctions of a σ-weight Maass Laplacian on the Poincaré disk, which are associated with the eigenvalue 4m (σ - 1 - m), m ∈ Z+ ∩ [0, (σ - 1) /2]. For each nonzero m, the associated coherent states transform constitutes the m-true-polyanalytic extension of a relativistic version of the second Bargmann transform, whose integral kernel is expressed in terms of a special Appel-Kampé de Fériet's hypergeometric function. The obtained results could be used to extend the known semi-classical analysis of quantum dynamics of the relativistic isotonic oscillator. [ABSTRACT FROM AUTHOR]

We examine the analytical structure of the nonlinear Liénard oscillator and show that it is a bi-Hamiltonian system depending upon the choice of the coupling parameters. While one has been recently studied in the context of a quantized momentumdependent mass system, the other Hamiltonian also reflects a similar feature in the mass function and also depicts an isotonic character. We solve for such a Hamiltonian and give the complete solution in terms of a confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

Two-loop massive Feynman integrals for ɸ4 field-theoretical model with long-range correlated disorder are considered. Massive integrals for the vertex function T(4) including two or three massless propagators for generic space dimension and for any value of the correlation parameter are evaluated analytically applying Mellin-Barnes method as well as familiar representation for one-loop integrals. Obtained expressions are presented in the form of hypergeometric functions. [ABSTRACT FROM AUTHOR]

Useful expressions of the derivatives, to any order, of Pochhammer and reciprocal Pochhammer symbolswith respect to their arguments are presented. They are building blocks of a procedure, recently suggested, for obtaining the e-expansion of functions of the hypergeometric class related to Feynman integrals. The procedure is applied to some examples of such kind of functions taken from the literature. [ABSTRACT FROM AUTHOR]

Partial wave expansion of the Coulomb-distorted plane wave is determined and studied. Dominant and sub-dominant asymptotic expansion terms are given and leading order three-dimensional asymptotic form is derived. The generalized hypergeometric function 2F2(a, a; a + l + 1, a - l; z) is expressed with the help of confluent hypergeometric functions and the asymptotic expansion of 2F2(a, a; a + l + 1, a - l; z) is simplified. [ABSTRACT FROM AUTHOR]

We consider the hypergeometric equation (1 - t)∂t∂t∂g + x3g = 0, whose unique analytic solution φ1(t; x) = 1 + O(t) near t = 0 for t = 1 becomes a generating function for multiple zeta values φ1(1; x) = f3(x) = 1 - ζ(3)x3 + ζ(3, 3)x6 - .... We apply the so-called WKB method to study solutions of the hypergeometric equation for large x and we calculate corresponding Stokes matrices. We prove that the function f3(x) near x = ∞ is also expressed via WKB type functions which subject to some Stokes phenomenon. This implies that f3(x) satisfies a sixth order linear differential equation with irregular singularity at infinity. [ABSTRACT FROM AUTHOR]

We consider Schrödinger equations for the quantum Painlevé equations. We present hypergeometric solutions of the Schrödinger equations for the quantum Painlevé equations, as particular solutions. We also give a representation theoretic correspondence between Hamiltonians of the Schrödinger equations for the quantum Painlevé equations and those of the Knizhnik-Zamolodchikov (KZ) equation or the confluent KZ equations. [ABSTRACT FROM AUTHOR]

Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure, and the theory of Frobenius manifolds. In 1 + 1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2 + 1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions. [ABSTRACT FROM AUTHOR]

The generalization of the factorization method performed by Mielnik [J. Math. Phys. 25, 3387 (1984)] opened new ways to generate exactly solvable potentials in quantum mechanics. We present an application of Mielnik's method to hypergeometric type operators. It is based on some solvable Riccati equations and leads to a unitary description of the quantum systems exactly solvable in terms of orthogonal polynomials or associated special functions. [ABSTRACT FROM AUTHOR]

In the investigation of two-body Coulomb Schrödinger equations with some types of nonhomogeneities, the particular solution can be expressed in terms of a two-variable Kampé de Fériet hypergeometric function. The asymptotic limit of the latter-for both variables being large but their ratio being a bound constant-is required in order to extract relevant physical information from the solutions. In this report the mathematical limit is provided. For that purpose, a particular series representation of the hypergeometric function-in terms of products of Kummer and Gauss functions-is first derived. [ABSTRACT FROM AUTHOR]

We solve the linear Ginzburg-Landau (GL) equation in the presence of a uniform magnetic field with cylindrical symmetry and we find analytic expressions for the eigenfunctions in terms of the confluent hypergeometric functions. The discrete spectrum results from an implicit equation associated to the boundary conditions and it is resolved in analytic form using the continued fractions formalism. We study the dependence of the spectrum and the eigenfunctions on the sample size and the surface conditions for solid and hollow cylindrical superconductors. Finally, the solutions of the nonlinear GL formalism are constructed as expansions in the linear GL eigenfunction basis and selected by minimization of the free energy. We present examples of vortex states and their energies for different samples in enhancing/suppressing superconductivity surroundings. [ABSTRACT FROM AUTHOR]

We study the gravitational behavior of a spherically symmetric radiating star when the fluid particles are in geodesic motion. We transform the governing equation into a simpler form which allows for a general analytic treatment. We find that Bernoulli, Ricatti, and confluent hypergeometric equations are possible. These admit solutions in terms of elementary functions and special functions. Particular models contain the Minkowski space-time and the Friedmann dust space-time as limiting cases. Our infinite family of solutions contains specific models found previously. For a particular metric we briefly investigate the physical features, derive the temperature profiles, and plot the behavior of the casual and acasual temperatures. [ABSTRACT FROM AUTHOR]

We investigate a basic question of analytic information theory, namely, the evaluation of the Fisher information and the relative Fisher information with respect to a non-negative function, for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second-order differential equations. We obtain explicit expressions for these information-theoretic properties via the expectation values of the coefficients of the differential equation. We illustrate our approach for various nonrelativistic D-dimensional wavefunctions and some special functions of physicomathematical interest. Emphasis is made in the Nikiforov–Uvarov hypergeometric-type functions, which include and generalize the Hermite functions and the Gauss and Kummer hypergeometric functions, among others. [ABSTRACT FROM AUTHOR]

This paper examines some solutions for confluent and double-confluent Heun equations. In the first place, we review two Leaver’s solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation [Leaver, E. W., J. Math. Phys. 27, 1238 (1986)] and introduce an additional expansion in series of irregular confluent hypergeometric functions. Then, we find the conditions under which one of these solutions can be written as a linear combination of the others. In the second place, by means of limiting procedures we generate solutions for the double-confluent equation as well as for special limits of both the confluent and double-confluent equations. Finally, we present some problems which are ruled by one or the other of these four equations. [ABSTRACT FROM AUTHOR]

The derivatives to any order of the confluent hypergeometric (Kummer) function F=1F1(a,b,z) with respect to the parameter a or b are investigated and expressed in terms of generalizations of multivariable Kampé de Fériet functions. Various properties (reduction formulas, recurrence relations, particular cases, and series and integral representations) of the defined hypergeometric functions are given. Finally, an application to the two-body Coulomb problem is presented: the derivatives of F with respect to a are used to write the scattering wave function as a power series of the Sommerfeld parameter. [ABSTRACT FROM AUTHOR]

The Dirac equation is solved in the rotating Bertotti–Robinson spacetime. The set of equations representing the Dirac equation in the Newman–Penrose formalism is decoupled into an axial and an angular part. The axial equation, which is independent of mass, is exactly solved in terms of hypergeometric functions. The angular equation is considered both for massless (neutrino) and massive spin-12 particles. For the neutrinos, it is shown that the angular equation admits an exact solution in terms of the confluent Heun equation. In the existence of mass, the angular equation does not allow an analytical solution, however, it is expressible as a set of first order differential equations apt for a numerical study. [ABSTRACT FROM AUTHOR]

The Dirac equation written on the boundary of the Nutku helicoid space consists of a system of ordinary differential equations. We tried to analyze this system and we found that it has a higher singularity than those of the Heun equations which give the solutions of the Dirac equation in the bulk. We also lose an independent integral of motion on the boundary. This facts explain why we could not find the solution of the system on the boundary in terms of known functions. We make the stability analysis of the helicoid and catenoid cases and end up with an Appendix which gives a new example wherein one encounters a form of the Heun equation. [ABSTRACT FROM AUTHOR]

In a recently published paper in this journal [A. Cheaghlou and O. Faizy, J. Math. Phys. 49, 022104 (2008)], the authors introduce the Gazeau–Klauder coherent states for the trigonometric Rosen–Morse potential as an infinite superposition of the wavefunctions. It is shown that their proposed measure to realize the resolution of the identity condition is not positive definite. Consequently, the claimed coherencies for the trigonometric Rosen–Morse wavefunctions cannot actually exist. [ABSTRACT FROM AUTHOR]

We suggest a systematic method of extension of quasiexactly solvable (QES) systems. We construct finite-dimensional subspaces on the basis of special functions (hypergeometric, Airy, Bessel ones) invariant with respect to the action of differential operators of the second order with polynomial coefficients. As an example of physical applications, we show that the known two-photon Rabi Hamiltonian becomes quasiexactly solvable at certain values of parameters when it can be expressed in terms of corresponding QES operators related to the hypergeometric function. [ABSTRACT FROM AUTHOR]

New explicit expressions are derived for the one-loop two-point Feynman integral with arbitrary external momentum and masses m12 and m22 in D dimensions. The results are given in terms of Appell functions, manifestly symmetric with respect to the masses mi2. Equating our expressions with previously known results in terms of Gauss hypergeometric functions yields reduction relations for the involved Appell functions that are apparently new mathematical results. [ABSTRACT FROM AUTHOR]

A nonlinear model representing the quantum harmonic oscillator on the sphere and the hyperbolic plane is solved in polar coordinates (r,[lowercase_phi_synonym]) by making use of a curvature-dependent formalism. The curvature κ is considered as a parameter and then the radial Schrödinger equation becomes a κ-dependent Gauss hypergeometric equation. The energy spectrum and the wave functions are exactly obtained in both the sphere S2 (κ>0) and the hyperbolic plane H2 (κ<0). A comparative study between the spherical and the hyperbolic quantum results is presented. [ABSTRACT FROM AUTHOR]

We find exact solutions of the Dirac equation in the 2+1 dimensional curved background by separation of variables. These solutions are given in terms of hypergeometric functions. We also perform the Gordon decomposition for the Dirac current to discuss the time dependence of the polarization densities and the magnetization density, and to show that the polarization densities are more effective than the magnetization density in the pair production in finite time intervals. [ABSTRACT FROM AUTHOR]

We derive the exact analytic form for the second-order positron-electron interaction term in the Faddeev three-body approach which is applicable in the nonrelativistic high energy region. Although there is no nonintegrable singularity in the six-dimensional integral form of this amplitude, here the basic difficulty arises from the presence of complex nonintegral exponents in the components included in the integrand. Consequently, three brunch cuts must be handled simultaneously. However, by using an integral representation of the gamma function, these brunch cuts are removed from the integrand. Expanding the radial parts of the initial and final wave functions further reduces the second-order positron-electron interaction term to a one-variable integral in terms of Bessel functions of the third kind. The different final closed expressions are ultimately derived in terms of the generalized hypergeometric functions for different regions of the scattering angle. [ABSTRACT FROM AUTHOR]

We show that second-order superintegrable systems in two-dimensional and three-dimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schrödinger equation can be expressed in terms of hypergeometric functions mFn and is QES if the Schrödinger equation admits polynomial solutions with coefficients necessarily satisfying a three-term or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze’s tenth-order polynomial QES problem in one set of separable coordinates and also leads to a fourth-order polynomial QES problem in another separable coordinate set. [ABSTRACT FROM AUTHOR]

We find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation (this is also known as confluent Heun equation). One solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable z, while the other is given by a series of modified Bessel functions and converges for |z|>|z0|, where z0 denotes a regular singularity. For short, the preceding limit is called Ince’s limit after Ince who have used the same procedure to get the Mathieu equations from the Whittaker-Hill ones. We find as well that, when z0 tends to zero, the Ince limit of the generalized spheroidal wave equation turns out to be the Ince limit of a double-confluent Heun equation, for which solutions are provided. Finally, we show that the Schrödinger equation for inverse fourth- and sixth-power potentials reduces to peculiar cases of the double-confluent Heun equation and its Ince’s limit, respectively. [ABSTRACT FROM AUTHOR]

We consider sums over the square lattice which depend only on radial distance, and provide formulas which enable sums of functions with Neumann series to be reexpressed as combinations of hypergeometric series. We illustrate the procedure using trigonometric sums previously studied by Borwein and Borwein, sums combining logarithms, Bessel functions Jλ, and powers of distance, and sums of Neumann functions. We also exhibit sums which may be evaluated analytically and recurrence formulas linking sums. [ABSTRACT FROM AUTHOR]

An eigenfunction expansion method involving hypergeometric functions is used to solve the partial differential equation governing the transport of radiation in an x-ray pulsar accretion column containing a radiative shock. The procedure yields the exact solution for the Green’s function, which describes the scattering of monochromatic radiation injected into the column from a source located near the surface of the star. Collisions between the injected photons and the infalling electrons cause the radiation to gain energy as it diffuses through the gas and gradually escapes by passing through the walls of the column. The presence of the shock enhances the energization of the radiation and creates a power-law spectrum at high energies, which is typical for a Fermi process. The analytical solution for the Green’s function provides important physical insight into the spectral formation process in x-ray pulsars, and it also has direct relevance for the interpretation of spectral data for these sources. Additional interesting mathematical aspects of the problem include the establishment of a closed-form expression for the quadratic normalization integrals of the orthogonal eigenfunctions, and the derivation of a new summation formula involving products of hypergeometric functions. By taking various limits of the general expressions, we also develop new linear and bilinear generating functions for the Jacobi polynomials. [ABSTRACT FROM AUTHOR]

Universal Drinfeld twists are inner automorphisms which relate the coproduct of a quantum enveloping algebra to the coproduct of the undeformed enveloping algebra. Even though they govern the deformation theory of classical symmetries and have appeared in numerous applications, no twist for a semisimple quantum enveloping algebra has ever been computed. It is argued that universal twists can be reconstructed from their well-known representations. A method to reconstruct an arbitrary element of the enveloping algebra from its irreducible representations is developed. For the twist this yields an algebra valued generating function to all orders in the deformation parameter, expressed by a combination of basic and ordinary hypergeometric functions. It is shown how the generating function can be expanded to the formal power series of the twist. An explicit expression for the universal twist of su(2) is given up to third order. [ABSTRACT FROM AUTHOR]

By confluencing a subset of upper and lower parameters in the generalized hypergeometric function [sub P]F[sub Q](a[sub 1,],...,a[sub P],c[sub 1],...,c[sub Q];z) with the variable z one obtains a lower-order hypergeometric function in the limit when the confluence parameters go to infinity. It is shown that this function is the first term in a convergent expansion in terms of functions of the same type with parameters increasing stepwise by integers ν and coefficients which are polynomials in the reciprocals of the confluence parameters. These polynomials have nonvanishing lowest degree terms whose power increases with ν. The expansion can hence be used to derive asymptotic expansions for large but finite absolute values of the confluence parameters. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]