A Generalized Hypergeometric Function¶Satisfying Four Analytic Difference Equations¶of Askey--Wilson Type

The hypergeometric function ₂ F ₁ can be written in terms of a contour integral involving gamma functions. We generalize this (Barnes) representation by using a certain generalized gamma function as a building block. In this way we obtain a new ₂ F ₁-generalization with various symmetry features. We determine the analyticity properties of the R-function in all of its eight arguments, and show that it is a joint eigenfunction of four distinct Askey–Wilson type difference operators, two acting on v and two on . The Askey–Wilson polynomials can be obtained by a suitable discretization of v or . Received: 21 December 1998 / Accepted: 14 April 1999     

NONCOMMUTATIVE HYPERGEOMETRIC AND BASIC HYPERGEOMETRIC EQUATIONS

Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100(14) (2003) 8138–8141] considered a matrix-valued analogue of the ₂F₁ Gauß hypergeometric function and showed that it is the unique solution of a matrix-valued hypergeometric equation analytic at z = 0 with value I, the identity matrix, at z = 0. We give an independent proof of Tirao's result, extended to the more general setting of hypergeometric functions over an abstract unital Banach algebra. We provide a similar (but more complicated-looking) result for a second type of noncommutative ₂F₁ Gauß hypergeometric function. We further give q-analogues for both types of noncommutative hypergeometric equations.     

Further extended Caputo fractional derivative operator and its applications

In this paper, our principle aim is to establish a new extension of the Caputo fractional derivative operator involving the generalized hypergeometric type function F _p (a, b; c; z; k), introduced by Lee et al. Some extensions of the generalized hypergeometric functions and their integral representations are also presented. Furthermore, linear and bilinear generating relations for the extended hypergeometric functions are obtained. We also present some properties of the extended fractional derivative operator.     

A unified treatment of the groups SO(4) and SO(3,1)

The irreducible representations of the group SO(4) in which the SO(3) subgroup is reduced are studied by an explicit construction of the operators and the basis in the spinor representation. The basis function which is formally identical with that for the coupling of two angular momentaj ₁ andj ₂ is expressible in terms of a hypergeometric function and strongly resembles the one for the irreducible representations of the groups SO(3,1). For the Lorentz group, the bases for the unitary representations which require unphysical values ofj ₁ andj ₂ are found to be analytic continuation of those for SO(4). The realization of the unitary irreducible representations of the group SO(4) in the Hilbert space of these functions leads, for appropriate unphysical values ofj ₁,j ₂, to the Gelfand-Naimark formula for the principal and complementary series of the representations of SO(3;1). The matrix elements for finite transformations of SO(4) and SO(3,1) can be evaluated, in this approach, in a unified manner by using standard properties of the hypergeometric function. These turn out to be a finite sum of₃ F ₂-functions which, as expected, are polynomials for SO(4) and infinite series for SO(3,1). A number of special matrix elements are calculated from the general formula and these agree with the results obtained previously.The authors are deeply indebted to Professor S.Dutta Majumdar fo many important suggestions and clarifications.     

On the equivalence of the Euclidean and Wightman formulation of field theory

A mistake in the paper [1] on the “Axioms for Euclidean Green's Functions” is corrected in the following sense: thanks to these axioms the Euclidean Schwinger functionsS _n can be analytically continued to the corresponding Wightman functionsW _n possessing all the correct analyticity properties and satisfying a generalized positivity condition in the complex domain. It is however suggested by the proof that their tempered behaviour near the Minkowski points cannot be guaranteed without additional assumptions.     

Properties of Generalized Univariate Hypergeometric Functions

Based on Spiridonov’s analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E ₇ (elliptic, hyperbolic) and of type E ₆ (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars’ relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions.     

Orbital currents and charge density waves in a generalized Hubbard ladder

We study a generalized Hubbard model on the two-leg ladder at zero temperature, focusing on a parameter region with staggered flux (SF)/d-density wave (DDW) order. To guide our numerical calculations, we first investigate the location of a SF/DDW phase in the phase diagram of the half-filled weakly interacting ladder using a perturbative renormalization group (RG) and bosonization approach. For hole doping δ away from half-filling, finite-system density-matrix renormalization-group (DMRG) calculations are used to study ladders with up to 200 rungs for intermediate-strength interactions. In the doped SF/DDW phase, the staggered rung current and the rung electron density both show periodic spatial oscillations, with characteristic wavelengths 2/δ and 1/δ, respectively, corresponding to ordering wavevectors 2k_F and 4k_F for the currents and densities, where 2k_F = π (1 − δ). The density minima are located at the anti-phase domain walls of the staggered current. For sufficiently large dopings, SF/DDW order is suppressed. The rung density modulation also exists in neighboring phases where currents decay exponentially. We show that most of the DMRG results can be qualitatively understood from weak-coupling RG/bosonization arguments. However, while these arguments seem to suggest a crossover from non-decaying correlations to power-law decay at a length scale of order 1/δ, the DMRG results are consistent with a true long-range order scenario for the currents and densities.     

Improved calculation of the structure functions of the nucleon in the O(4, 2)-infinite multiplet model

A new investigation of the inelastic electon scattering from proton in the O(4, 2) models is presented. The resultant explicit structure functions in the limit satisfy scaling, F₁(ξ) ≠ 0 (σ_T ≠ 0), the Drell-Yan relation F₂(ξ) ～ (1?ξ)³ and, approximately, the Callan-Gross relation F₂(ξ) ≈ 2ξF₁(ξ).     

Some families of generating functions and associated hypergeometric transformation and reduction formulas

Summation, transformation and reduction formulas for various families of hypergeometric functions in one, two and more variables are potentially useful in many diverse areas of applications. The main object of this paper is to derive several substantially more general results on this subject than those considered recently by Neethu et al. [7] in connection with Bailey’s transformation involving the Gauss hypergeometrc function 2F ₁ (see [1]). The methodology used here is based essentially on some families of hypergeometric generating functions. Relevant connections of the results presented in this paper with those in the earlier works are also pointed out.     

Identities for Generalized Appell Functions and the Blow-up Formula

In this paper, we prove identities for a class of generalized Appell functions which are based on the \({{\rm A}_2}\) root lattice. The identities are reminiscent of periodicity relations for the classical Appell function and are proven using only analytical properties of the functions. Moreover, they are a consequence of the blow-up formula for generating functions of invariants of moduli spaces of semi-stable sheaves of rank 3 on rational surfaces. Our proof confirms that in the latter context, different routes to compute the generating function (using the blow-up formula and wall-crossing) do arrive at identical q-series. The proof also gives a clear procedure on how to prove analogous identities for generalized Appell functions appearing in generating functions for sheaves with rank \({r>3}\).     

The geometry of Ingarden spaces

The Randers spaces RFⁿ were introduced by R. S. Ingarden. They are considered as Finsler spaces Fⁿ = (M, α + β) equipped with the Cartan nonlinear connection. In the present paper we define and study what we call the Ingarden spaces, I Fⁿ, as Finsler spaces I Fⁿ = (M, α + β) equipped with the Lorentz nonlinear connection. The spaces R Fⁿ and I Fⁿ are completely different. For I Fⁿ we discuss: the variational problem, Lorentz nonlinear connection, canonical N-metrical connection and its structure equations, the Cartan 1-form ω, the electromagnetic 2-form tF and the almost symplectic 2-form 0. The formula dω = F+θ is established. It has as a consequence the generalized Maxwell equations. Finally, the almost Hermitian model of I Fⁿ is constructed.     

Difference equations in spin chains with a boundary

Correlation functions and form factors in vertex models or spin chains are known to satisfy certain difference equations called the quantum Knizhnik-Zamolodchikov equations. We find similar difference equations for the case of semi-infinite spin chain systems with integrable boundary conditions. We derive these equations using the properties of the vertex operators and the boundary vacuum state, or alternatively through corner transfer matrix arguments for the eight-vertex model with a boundary. The spontaneous boundary magnetization is found by solving such difference equations. The boundary S-matrix is also proposed and compared, in the sine-Gordon limit, with Ghoshal-Zamolodchikov's result. The axioms satisfied by the form factors in the boundary theory are formulated.     

General proof of Osterwalder-Schrader positivity for the Wilson action

We extend the proof by Osterwalder and Seiler of reflection positivity of lattice gauge theories with fermions to the case in which the reflection is performed with respect to planes containing sites. Our proof applies to all observables and relies on the particular structure of the Wilson action. This assures the positivity of the correlation functions (F)F for any gauge invariant functionF and at all euclidean time separations as of interest in Monte Carlo calculations.     

Radiative pion decay: Determination of FA(0) from τ-lepton decay data

The form factor F_A(0) in π→eνγ is calculated, at the soft pion point, directly from data on the semileptonic τ-lepton decays τ→ν_τ + nπ, which determine the relevant vector and axial-vector hadronic spectral functions. The reliability of this calculation is assessed by using these fitted spectral functions to check the saturation of the first two Weinberg sum rules. The results are: F_A(0)=0.017±0.004, and γ(0)F_A(0)/F_V(0)=0.67±0.04±0.16.     

Generalized Sturm expansions of the Coulomb Green’s function and two-photon Gordon formulas

The radial component of the Coulomb Green’s function (CGF) is written in the form of a double series in Laguerre polynomials (Sturm’s functions in the Coulomb problem), which contains two free parameters α and α′. The obtained result is applicable both in the nonrelativistic case and for the CGF of the squared Dirac equation with a Coulomb potential. The CGF is decomposed into the resonance and potential components (the latter is a smooth function of energy) for α = α′. In the momentum representation, the CGF with the free parameters is written in the form of an expansion in four-dimensional spherical functions. The choice of the parameters α and α ′ in accordance with the specific features of the given problem radically simplifies the calculation of the composite matrix elements for electromagnetic transitions. Closed analytic expressions (in terms of hypergeometric functions) are obtained for the amplitudes of bound-bound and bound-free two-photon transitions in the hydrogen atom from an arbitrary initial state ¦nl〉, which generalize the known (one-photon) Gordon formulas. The dynamic polarizability tensor components α_nlm(ω) for an arbitrary n are expressed in terms of the hypergeometric function ₂ F ₁ depending only on l and $\tilde \omega $ and through the polynomial functions $f_{nl} (\tilde \omega )$ of frequency $\tilde \omega = {{\hbar \omega } \mathord{\left/ {\vphantom {{\hbar \omega } {\left| {E_n } \right|}}} \right. \kern-0em} {\left| {E_n } \right|}}$ . The Rydberg (n ? 1) and threshold (?ω ～ ¦ E _n¦) asymptotic forms of polarizabilities are investigated.     

Modular Hypergeometric Residue Sums of Elliptic Selberg Integrals

It is shown that the residue expansion of an elliptic Selberg integral gives rise to an integral representation for a multiple modular hypergeometric series. A conjectural evaluation formula for the integral then implies a closed summation formula for the series, generalizing both the multiple basic hypergeometric ₈₇ sum of Milne-Gustafson type and the (one-dimensional) modular hypergeometric ₈₇ sum of Frenkel and Turaev. Independently, the modular invariance ensures the asymptotic correctness of our multiple modular hypergeometric summation formula for low orders in a modular parameter.     

Integrable Structure of Ginibre’s Ensemble of Real Random Matrices and a Pfaffian Integration Theorem

In the recent publication (E. Kanzieper and G. Akemann in Phys. Rev. Lett. 95:230201, 2005), an exact solution was reported for the probability p _n,k to find exactly k real eigenvalues in the spectrum of an n×n real asymmetric matrix drawn at random from Ginibre’s Orthogonal Ensemble (GinOE). In the present paper, we offer a detailed derivation of the above result by concentrating on the proof of the Pfaffian integration theorem, the key ingredient of our analysis of the statistics of real eigenvalues in the GinOE. We also initiate a study of the correlations of complex eigenvalues and derive a formula for the joint probability density function of all complex eigenvalues of a GinOE matrix restricted to have exactly k real eigenvalues. In the particular case of k=0, all correlation functions of complex eigenvalues are determined.     

Numerical and asymptotic scattering from two inhomogeneous lenses

The scattering of a time-harmonic, plane electromagnetic wave by either a Maxwell fish-eye or a metal-like lens is considered. Numerical values of the monostatic cross sections are obtained by evaluating the exact, infinite series solutions of hypergeometric functions of the₂ F ₁ type. These results are compared with the high-frequency asymptotic estimates based on either geometrical optics or a modified Watson transformation.     

Lifshitz integral in closed forms

Closed forms of them-fold Lifshitz integral are given in terms of the hypergeometric function₂ F ₁ and in its arguments. The applicability of each form is discussed.On Sabbatical leave at Institute of Theoretical Physics, Johannes Kepler University, Linz, A-4040 Linz-Auhof, Austria.