DOUBLE INTEGRALS INVOLVING PRODUCT OF TWO GENERALIZED HYPERGEOMETRIC FUNCTIONS

Summary In this note a problem on exact moments of order statistics from a power-function distribution is considered. The characteristic function of the kth order statistic is obtained and moments about the origin of the kth order statistic are expressed in terms of gamma functions. An exact expression for the covariance of any two order statistics Yi < Yj is obtained in terms of beta and gamma functions. Various recurrence relations between the expected values of order statistics are also obtained.

In this paper, one hundred interesting double integrals involving Gauss's hypergeometric function in the form of four general integrals (twenty five each) have been evaluated in terms of gamma function. More than two hundred special cases have also been given.

We present new asymptotic expansions of the logarithm of the gamma function in terms of the polygamma functions. Based on these expansions, we prove new complete mono- tonicity properties of some functions involving the gamma and polygamma functions. As con- sequences of them we establish new upper and lower bounds for the gamma function in terms of the polygamma functions.

In this research note, an interesting integral involving hypergeometric function has been evaluated in terms of gamma function. It is further used to evaluate an integral involving product of two generalized hypergeometric functions. A few very interesting special cases have also been given

We evaluate all phase space master integrals which are required for the total cross section of generic 2 -> 1 processes at NNLO as a series expansion in the dimensional regulator epsilon. Away from the limit of threshold production, our expansion includes one order higher than what has been available in the literature. At threshold, we provide expressions which are valid to all orders in terms of Gamma functions and hypergeometric functions. These results are a necessary ingredient for the renormalization and mass factorization of singularities in 2 -> 1 inclusive cross sections at NNNLO in QCD.

Euler discovered the gamma function, Γ( x ), when he extended the domain of the factorial function. Thus Γ( x ) is a meromorphic function equal to ( x − 1)! when x is a positive integer. The gamma function has several representations, but the two most important, found by Euler, represent it as an infinite integral and as a limit of a finite product. We take the second as the definition. Instead of viewing the beta function as a function, it is more illuminating to think of it as a class of integrals – integrals that can be evaluated in terms of gamma functions. We therefore often refer to beta functions as beta integrals. In this chapter, we develop some elementary properties of the beta and gamma functions. We give more than one proof for some results. Often, one proof generalizes and others do not. We briefly discuss the finite field analogs of the gamma and beta functions. These are called Gauss and Jacobi sums and are important in number theory. We show how they can be used to prove Fermat's theorem that a prime of the form 4 n + 1 is expressible as a sum of two squares. We also treat a simple multidimensional extension of a beta integral, due to Dirichlet, from which the volume of an n -dimensional ellipsoid can be deduced.

In this paper, we evaluate archimedean zeta integrals for automorphic L-functions on GL n × GL n-1+l and on SO2n+1 × GL n+l , for l = −1, 0, and 1. In each of these cases, the zeta integrals in question may be expressed as Mellin transforms of products of class one Whittaker functions. Here, we obtain explicit expressions for these Mellin transforms in terms of Gamma functions and Barnes integrals. When l = 0 or l = 1, the archimedean zeta integrals amount to integrals over the full torus. We show that, as has been predicted by Bump for such domains of integration, these zeta integrals are equal to the corresponding local L-factors—which are simple rational combinations of Gamma functions. (In the cases of GL n × GL n-1 and GL n × GL n this has, in large part, been shown previously by the second author of the present work, though the results here are more general in that they do not require the assumption of trivial central characters. Our techniques here are also quite different. New formulas for GL(n, R) Whittaker functions, obtained recently by the authors of this work, allow for substantially simplified computations). In the case l = −1, we express our archimedean zeta integrals explicitly in terms of Gamma functions and certain Barnes-type integrals. These evaluations rely on new recursive formulas, derived herein, for GL(n, R) Whittaker functions. Finally, we indicate an approach to certain unramified calculations, on SO2n+1 × GL n and SO2n+1 × GL n+1, that parallels our method herein for the corresponding archimedean situation. While the unramified theory has already been treated using more direct methods, we hope that the connections evoked herein might facilitate future archimedean computations.

The first two functions discussed in this chapter are due to Euler. The third is usually associated with Riemann, although it was also studied earlier by Euler. Collectively they are of great importance historically, theoretically, and for purposes of calculation. Historically and theoretically, investigation of these functions and their properties has provided considerable impetus to the study and understanding of fundamental aspects of mathematical analysis, including limits, infinite products, and analytic continuation. They have also motivated advances in complex function theory, such as the theorems of Weierstrass and Mittag-Leffler on representations of entire and meromorphic functions. The zeta function and its generalizations are intimately connected with questions of number theory. From the point of view of calculation, many of the explicit constants of mathematical analysis, especially those that come from definite integrals, can be evaluated in terms of the gamma and beta functions. There is much to be said for proceeding historically in discussing these and other special functions, but we shall not make a point of doing so. In mathematics it is often, even usually, the case that later developments cast a new light on earlier ones. One result is that later expositions can often be made both more efficient and, one hopes, more transparent than the original derivations. After introducing the gamma and beta function and their basic properties, we turn to a number of important identities and representations of the gamma function and its reciprocal. Two characterizations of the gamma function are established, one based on complex analytic properties, the other based on a geometric property. Asymptotic properties of the gamma function are considered in detail. The psi function and the incomplete gamma function are introduced. The identity that evaluates the beta integral in terms of gamma functions has important modern generalizations due to Selberg and Aomoto. Aomoto's proof is sketched. The zeta function, its functional equation, and Euler's evaluation of ζ ( n ) for n = 2, 4, 6, …, are the subject of the final section. The gamma and beta functions The gamma function was introduced by Euler in 1729 [119] in answer to the question of finding a function that takes the value n ! at each nonnegative integer n . At that time, a “function” was understood as a formula expressed in terms of the standard operations of algebra and calculus, so the problem was not trivial.

A multivariable hypergeometric function which was studied recently by Niukkanen (1984) and Srivastava (1985), provides an interesting and useful unification of the generalised hypergeometric pFq function of one variable (with p numerator and q denominator parameters), Appell and Kampe de Feriet's hypergeometric functions of two variables, and Lauricella's hypergeometric functions of n variables, and also of many other classes of hypergeometric series which arise naturally in various physical and quantum chemical applications. Indeed, as already observed by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalised Lauricella hypergeometric function of n variables, which was first introduced and studied systematically by Srivastava and Daoust (1969). By employing such useful connections of this function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely, Srivastava presented several interesting and useful properties of this multivariable hypergeometric function, most of which did not appear in the work of Niukkanen. The object of this sequel to Srivastava's work is to derive a number of new Neumann expansions in series of Bessel functions for the multivariable hypergeometric function from substantially more general expansions involving, for example, multiple series with essentially arbitrary terms. Some interesting special cases of the Neumann expansions presented here are also indicated.

The generalized Laguerre polynomials form a complete set (orthogonal and normalized) in the space ** with respect to a certain weighting function because they are the Eigen functions of a second-order differential operator. Here we shall show how to expand some well-known classical polynomials such as the Legendre and the Hermite polynomials in series of generalized Laguerre polynomials. Since the generalized Laguerre orthogonal polynomials are set of polynomials that are mutually orthogonal to each other with respect to a measure of weighting function that is just the integrand of the gamma function, thus it grants us the guarantee of the ability to expand the first kind of Bessel functions and the gamma function in terms of the generalized Laguerre polynomials. The series expansion in terms of the generalized Laguerre polynomials can be achieved by following various approaches. For instance, the series expansion of the first kind Bessel functions is gained by the generalized hypergeometric function approach, whereas the series expansion of gamma function was obtained directly by the usual way that is by calling the orthogonality and orthonormality properties of the generalized Laguerre polynomials. Another powerful technique to gain the series expansion of the classical polynomials is the generating function approach as it has been followed here to obtain a series expansion of the Legendre and the Hermite polynomials in terms of the generalized Laguerre polynomials. The series expansion in series of the generalized Laguerre polynomials has a variety of applications in mathematics, physics, and engineering

Estimating gamma function by digamma function

In a recent paper [On Some Unified Integrals, Advances in Comput. Math. and Its Applications, Vol. 1, No. 3, PP. 151-153 (2012)], the author has evaluated three very interesting integrals involving hypergeometric function in terms of gamma function. In this paper, three new unified integrals involving Fox’s H-function have been evaluated. By specializing the parameters, we can easily obtain a large number of new and known integrals including one obtained earlier by Garg and Mittal. The results established in this paper are simple, interesting, easily established and may be useful.

We solve the Klein–Gordon equation in the presence of the hyperbolic tangent potential. The scattering solutions are derived in terms of hypergeometric functions. The reflection, R, and transmission, T, coefficients are calculated in terms of gamma function and, superradiance is discussed, when the reflection coefficient, R, is greater than one.

K shell internal conversion coefficients at threshold

The longitudinal propagation and reflection of a plane electromagnetic wave in a horizontally stratified magneto-ionic medium is considered. In this case Maxwell's equations reduce to two uncoupled ordinary second-order differential equations, describing the propagation of two elliptically polarized plane waves. The electron density of the medium is assumed to vary with the vertical Cartesian coordinatez according to the Epstein law. Rigorous solutions of the relevant differential equations can be obtained either in the form of hypergeometric functions or in the form of an integral representation. The reflection coefficients of both waves are then expressed in terms of gamma functions. The following quantities are considered in detail in their dependence on the parameters involved: the modulus of the reflection coefficient, the phase delay time and the group delay time. Some numerical results are given.