Hypergeometric Function

We calculate perturbative Wilson loops of various sizes up to loop order n = 20 at different lattice sizes for pure plaquette and tree-level improved Symanzik gauge theories using the technique of Numerical Stochastic Perturbation Theory. This allows us to investigate the behavior of the perturbative series at high orders. We observe differences in the behavior of perturbative coefficients as a function of the loop order. Up to n = 20 we do not see evidence for the often assumed factorial growth of the coefficients. Based on the observed behavior we sum this series in a model with hypergeometric functions. Alternatively we estimate the series in boosted perturbation theory. Subtracting the estimated perturbative series for the average plaquette from the non-perturbative Monte Carlo result we estimate the gluon condensate.

Brizolis asked the question: does every prime p have a pair (g,h) such that h is a fixed point for the discrete logarithm with base g? The first author previously extended this question to ask about not only fixed points but also two-cycles, and gave heuristics (building on work of Zhang, Cobeli, Zaharescu, Campbell, and Pomerance) for estimating the number of such pairs given certain conditions on g and h. In this paper we extend these heuristics and prove results for some of them, building again on the aforementioned work. We also make some new conjectures and prove some average versions of the results.

The Digamma and Polygamma functions are important tools in mathematical physics, not only for its many properties but also for the applications in statistical mechanics and stellar evolution. In many textbooks is found its develop almost by the same procedure. In this paper expressions for the Digamma and Polygamma functions, in terms of hypergeometric functions, are found through the derivative definition.

We define a q-analog of the modified Bessel and Bessel-Macdonald functions. As for the q-Bessel functions of Jackson there is a couple of functions of the both kind. They are arisen in the Harmonic analysis on quantum symmetric spaces similarly to their classical counterpart. Their definition is based on the power expansions. We derive the recurrence relations, difference equations, q-Wronskians, and an analog of asymptotic expansions which turns out is exact in some domain if q = 1. Some relations for the basic hypergeometric function which follow from this fact are discussed.

The hypergeometric function of a real variable is computed for arbitrary real parameters. The transformation theory of the hypergeometric function is used to obtain rapidly convergent power series. The divergences that occur in the individual terms of the transformation for integer parameters are removed using a finite difference technique. ᮊ 1997 Academic Press 79

In his work on the elimination of the nodes in the Three-Body Problem, Jacobi considered a certain straight line (defined below, in the Introduction) which lies always within the invariable plane. Here we settle the question of Wintner on the classification of all motions for which this line fails to exist.

We deal here with a class of integral transformations with respect to parameters of hypergeometric functions or the index transforms. In particular, we treat the familiar Olevskii transform, which is associated with the Gauss hypergeometric function as a kernel. It involves, in turn, as particular cases index transforms of the Mehler-Fock type which are used in the mathematical theory of elasticity. Boundedness L2properties for the Olevskii transform are investigated. The Plancherel theorem is proved. It shows that the Olevskii transform is an isometric isomorphism between two weighted L2 -spaces. More examples of such isomorphisms are exhibited for the Mehler-Fock type transforms.

Symmetry and Separation of Variables: Encyclopedia of Mathematics and its Applications: Volume 4 Willard Miller Frontmatter More information vi Contents Chapter 3 The Three-Variable Helmholtz and Laplace Equations.. . 160 3.1 The Helmholtz Equation (A 3 + <o 2)^ = 0 160 3.2 A Hilbert Space Model: The Sphere S 2 169 3.3 Lame Polynomials and Functions on the Sphere 3.4 Expansion Formulas for Separable Solutions of the Helmoltz Equation 3.5 Non-Hilbert Space Models for Solutions of the Helmholtz Equation 3.6 The Laplace Equation A 3^ = 0 204 3.7 Identities Relating Separable Solutions of the Laplace Equation 213 Exercises 222 Chapter 4 The Wave Equation 223 4.1 The Equation * "-A 2 * = 0 223 4.2 The Laplace Operator on the Sphere 4.3 Diagonalization of P 0 , P 2 , and D 4.4 The Schrodinger and EPD Equations 4.5 The Wave Equation (3 //-A 3)^(x) = 0 Exercises Chapter 5 The Hypergeometric Function and Its Generalizations.. .

A new heuristic method for the evaluation of definite integrals is presented. This method of brackets has its origin in methods developed for the evaluation of Feynman diagrams. We describe the operational rules and illustrate the method with several examples. The method of brackets reduces the evaluation of a large class of definite integrals to the solution of a linear system of equations.

Hypergeometric functions are a generalization of exponential functions. They are explicit, computable functions that can also be manipulated analytically. The functions and series we use in quantitative economics are all special cases of them. In this paper, a unified approach to hypergeometric functions is given. As a result, some potentially useful general applications emerge in a number of areas such as in econometrics and economic theory. The greatest benefit from using these functions stems from the fact that they provide parsimonious explicit (and interpretable) solutions to a wide range of general problems.

Este artículo presenta una colección de funciones computacionales que son utilizadas en la implementación de un análisis bayesiano exhaustivo para la diferencia de dos proporciones. Con este fin, se discute la estimación puntual, la estimación mediante intervalos de credibilidad y la inferencia predictiva desde dos escenarios: el primero basado en las densidades exactas a priori y a posteriori (construidas mediante la primera función hipergeométrica de Appell) y el segundo basado en densidades simuladas (mediante un algoritmo de cadenas de Markov con métodos de Monte Carlo). La implementación de estas funciones se realiza en el programa estadístico R, porque es un software libre, funciona bien en múltiples plataformas y permite enmarcar estas funciones bajo un objeto computacional denominado "paquete". Palabras clave: estimación; funciones en R; inferencia bayesiana; proporciones. Clasificación JEL: C11; C12; C63. 2000MSC: 62C10; 62F03; 62P20; 90-08. Artículo recibido el 13 de octubre de 2009 y aceptado el 30 de noviembre de 2009.

The theory of theta functions is used to derive hypergeometric transformation formulas of degrees 3, 7, 11 and 23. As a consequence of the theory that is developed, some new series for 1/π are obtained that are similar to a class investigated by Ramanujan.

A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pull-back of one of the few standard hypergeometric equations with algebraic solutions. The most interesting cases are hypergeometric equations with tetrahedral, octahedral or icosahedral monodromy groups. We give an algorithm for computing Klein's pull-back coverings in these cases, based on certain explicit expressions (Darboux evaluations) of algebraic hypergeometric functions. The explicit expressions can be computed from a data base (covering the Schwarz table) and using contiguous relations. Klein's pull-back transformations also induce algebraic transformations between hypergeometric solutions and a standard hypergeometric function with the same finite monodromy group.

Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameter family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the "rational Cherednik algebra", and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups G(m, p, N ), the set of singular parameters in the parameter family of these structures is described explicitly, using the theory of nonsymmetric Jack polynomials.

The hyperbolic sup norm of the pre-Schwarzian derivative of a locally univalent function on the unit disk measures the deviation of the function from similarities. We present sharp norm estimates for the Alexander transforms of convex functions of order α, 0 α < 1.  2004 Elsevier Inc. All rights reserved.

Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the argument variable. The paper presents general elementary expressions of these dihedral hypergeometric functions, involving finite bivariate sums expressible as terminating Appell's F2 or F3 series. Additionally, trigonometric expressions for the dihedral functions are presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z) are considered.

We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the approach of obtaining iterated solutions to the differential equations associated with hypergeometric functions to prove the following result: Theorem 1. The epsilon-expansion of a generalized hypergeometric function with integer values of parameters, p F p−1 (I 1 + a 1 ε, . . . , I p + a p ε; I p+1 + b 1 ε, . . . , I 2p−1 + b p−1 ; z) , is expressible in terms of generalized polylogarithms with coefficients that are ratios of polynomials. The method used in this proof provides an efficient algorithm for calculating of the higherorder coefficients of Laurent expansion.

In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman-Opdam hypergeometric functions of type BC. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds G/K over one of the skew fields F = R, C, H. We write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters. In each of the three cases, we obtain a series of hypergroup algebras which include the commutative convolution algebras of K-biinvariant functions on G as special cases. The characters are given by the associated hypergeometric functions.

"In this article, we have derived the probability density functions of the productand the quotient of two independent random variables having Gauss hypergeometricdistribution. These densities have been expressed in terms of Appell'srst hypergeometric function F1. Further, Renyi and Shannon entropies havealso been derived for the Gauss hypergeometric distribution"

A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of products of three Bessel functions.

a b s t r a c t Let AðpÞ; p 2 N, be a class of functions f : f ðzÞ ¼ z p þ P 1 k¼pþn a k z k analytic in the open unit disc E. We use Carlson-Shaffer operator for p-valent functions to define and study certain classes of analytic functions. Inclusion results, a radius problem and some other interesting properties are discussed. . In (1.2), we use Pochhammer symbol ðkÞ k or the shifted factorial which is defined as

In this article, by the use of a further generalization of the algebraic method of separation of variables, the Dirac equation is separated in a family of space-times where it is not possible to find a complete set of first order commuting differential operators. After separation of variables, the Dirac equation is reduced to a set of coupled ordinary differential equations and some exact solutions corresponding to cosmological backgrounds and gravitational waves are computed in terms of hypergeometric functions. The Klein-Gordon equation in this background field is also discussed.

The main purpose of this paper is to present a number of potentially useful integral representations for the familiar Mathieu a-series as well as for its alternating version. These results are derived here from many different considerations and are shown to yield sharp bounding inequalities involving the Mathieu and alternating Mathieu a-series. Relationships of the Mathieu a-series with the Riemann Zeta function and the Dirichlet Eta function are also considered. Such special functions as the classical Bessel function J m (z) and the confluent hypergeometric functions 0 F 1 and 1 F 2 are characterized by means of certain Fredholm type integral equations of the first kind, which are associated with some of these Mathieu type series. Several integrals containing Mathieu type series are also evaluated. Finally, some closely-related new questions and open problems are indicated with a view to motivating further investigations on the subject of this paper.

This paper continues a study of one and two variable function space models of irreducible representations of q-analogs of Lie enveloping algebras, motivated by recurrence relations satis ed by q-hypergeometric functions. Here we consider the quantum algebra Uq (su 2 ). We show that various q-analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and we compute the corresponding matrix elements of the \group operators" on these representation spaces. This \local" approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. We show that the matrix elements themselves transform irreducibly under the action of the quantum algebra. We nd an alternate and simpler derivation of a q-analog, due to Groza, Kachurik and Klimyk, of the Burchnall-Chaundy formula for the product of two hypergeometric functions 2 F 1 . It is interpreted here as the expansion of the matrix elements of a \group operator" (via the exponential mapping) in a tensor product basis in terms of the matrix elements in a reduced basis.

A tetraparametric univariate distribution generated by the Gaussian hypergeometric function that includes the Waring and the generalized Waring distributions as particular cases is presented. This distribution is expressed as a generalized beta type I mixture of a negative binomial distribution, in such a way that the variance of the tetraparametric model can be split into three components: randomness, proneness and liability. These results are extensions of known analogous properties of the generalized Waring distribution. Two applications in the fields of sport and economy are included in order to illustrate the utility of the new distribution compared with the generalized Waring distribution.

We present explicit expressions of the Poisson kernels for geodesic balls in the higher dimensional spheres and real hyperbolic spaces. As a consequence, the Dirichlet problem for the projective space is explicitly solved. Comparison of different expressions for the same Poisson kernel lead to interesting identities concerning special functions.

We introduce the k-generalized gamma function Γ k , beta function B k and Pochhammer k-symbol (x) n,k . We prove several identities generalizing those satisfied by the classical gamma function, beta function and Pochhammer symbol. We provide integral representation for the Γ k and B k functions.

The main object of this paper is to introduce and investigate a class of analytic functions with negative coefficients defined by the Wright generalized hypergeometric function. By using the extreme points theory we obtain coefficient estimates and distortion theorems in the class of functions.

Performance analysis of equal-gain combining (EGC) diversity systems is notoriously difficult only more so given that the closed-form probability density function (pdf) of the EGC output is only available for dual-diversity combining in Rayleigh fading. In this paper, a powerful frequency-domain approach is therefore developed in which the average error-rate integral is transformed into the frequency domain, using Parseval's theorem. Such a transformation eliminates the need for computing (or approximating) the EGC output pdf (which is unknown), but instead requires the knowledge of the corresponding characteristic function (which is readily available). The frequency-domain method also circumvents the need to perform multiple-fold convolution integral operations, usually encountered in the calculation of the pdf of the sum of the received signal amplitudes. We then derive integral expressions for the average symbol-error rate of an arbitrary two-dimensional signaling scheme, with EGC reception in Rayleigh, Rician, Nakagami-, and Nakagamifading channels. For practically important cases of second-and third-order diversity systems in Nakagami fading, both coherent and noncoherent detection methods for binary signaling are analyzed using the Appell hypergeometric function. A number of closed-form solutions are derived in which the results put forward by Zhang are shown to be special cases.

In this paper, we derive conditions on the parameters a, b, c so that the function zF (a, b; c; z) is starlike in D, where F (a, b; c; z) denotes the classical hypergeometric function. We give some consequences of our results including some mapping properties of convolution operator.

The null geodesic equations that describe motion of photons in Kerr spacetime are solved exactly in the presence of the cosmological constant Λ. The exact solution for the deflection angle for generic light orbits (i.e. non-polar, non-equatorial) is calculated in terms of the generalized hypergeometric functions of Appell and Lauricella. We then consider the more involved issue in which the black hole acts as a 'gravitational lens'. The constructed Kerr black hole gravitational lens geometry consists of an observer and a source located far away and placed at arbitrary inclination with respect to black hole's equatorial plane. The resulting lens equations are solved elegantly in terms of Appell-Lauricella hypergeometric functions and the Weierstraß elliptic function. We then, systematically, apply our closed form solutions for calculating the image and source positions of generic photon orbits that solve the lens equations and reach an observer located at various values of the polar angle for various values of the Kerr parameter and the first integrals of motion. In this framework, the magnification factors for generic orbits are calculated in closed analytic form for the first time. The exercise is repeated with the appropriate modifications for the case of non-zero cosmological constant.

The 15 Gauss contiguous relations for 2F1 hypergeometric series imply that any three 2F1 series whose corresponding parameters differ by integers are linearly related (over the field of rational functions in the parameters). We prove several properties of coefficients of these general contiguous relations, and use the results to propose effective ways to compute contiguous relations. We also discuss contiguous relations of generalized and basic hypergeometric functions, and several applications of them.

The null geodesic equations that describe motion of photons in Kerr spacetime are solved exactly in the presence of the cosmological constant Λ. The exact solution for the deflection angle for generic light orbits (i.e. non-polar, non-equatorial) is calculated in terms of the generalized hypergeometric functions of Appell and Lauricella. We then consider the more involved issue in which the black hole acts as a `gravitational lens&#39;. The constructed Kerr black hole gravitational lens geometry consists of an observer and a source located far away and placed at arbitrary inclination with respect to the black hole&#39;s equatorial plane. The resulting lens equations are solved elegantly in terms of Appell-Lauricella hypergeometric functions and the Weierstrass elliptic function. We then, systematically, apply our closed form solutions for calculating the image and source positions of generic photon orbits that solve the lens equations and reach an observer located at various values ...

Throughout the present note we abbreviate the set of p parameters a1,…,ap by (ap), with similar interpretations for (bq), etc. Also, by [(ap)]m we mean the product , where [λ]m = Г(λ + m)/ Г(λ), and so on. One of the main results we give here is the expansion formula(1)which is valid, by analytic continuation, when, p,q,r,s,t and u are nonnegative integers such that p+r &lt; q+s+l (or p+r = q+s+l and |zω| &lt;1), p+t &lt; q+u (or p + t = q + u and |z| &lt; 1), and the various parameters including μ are so restricted that each side of equation (1) has a meaning.

We discuss how the derivative of solutions of some Heun's differential equations can be given, in some particular cases, from the solution of another Heun's equation. This work uses some algebraic links between Heun and hypergeometric equations.

This paper is an attempt at studying the neoclassical Solow-Swan model within a framework where the change over time of the labor-force is given by the logistic population model. In the canonical Solow-Swan model, the growth rate of population is constant, yielding an exponential behavior of population size over time, which is clearly unrealistic and unsustainable in the very long-run. A more realistic approach would be to consider a logistic law for the population growth rate. In this framework, the model is proved to have a unique equilibrium (a node), which is globally asymptotically stable, and its solution is shown to have a closed-form expression via Hypergeometric functions.