Barnes Integrals Research Articles

On differentiation with respect to parameters of the functions of the Mittag–Leffler type

The formal term-by-term differentiation with respect to parameters is demonstrated to be legitimate for the Mittag–Leffler-type functions. The justification of differentiation formulas is made by using the concept of the uniform convergence. This approach is applied to the Mittag–Leffler function depending on two parameters and, additionally, for the three-parametric Mittag–Leffler functions (namely, for the Prabhakar function and the Le Roy-type function), as well as for the four-parametric Mittag–Leffler function (and, in particular, for the Wright function). The differentiation with respect to the involved parameters is discussed also in case those special functions which are represented via the Mellin–Barnes integrals.

Hadronic Light-by-Light Corrections to the Muon Anomalous Magnetic Moment

We review the hadronic light-by-light (HLbL) contribution to the muon anomalous magnetic moment. Upcoming measurements will reduce the experimental uncertainty of this observable by a factor of four; therefore, the theoretical precision must improve accordingly to fully harness such an experimental breakthrough. With regards to the HLbL contribution, this implies a study of the high-energy intermediate states that are neglected in dispersive estimates. We focus on the maximally symmetric high-energy regime and in-quark loop approximation of perturbation theory, following the method of the OPE with background fields proposed by Bijnens et al. in 2019 and 2020. We confirm their results regarding the contributions to the muon g−2. For this, we use an alternative computational method based on a reduction in the full quark loop amplitude, instead of projecting on a supposedly complete system of tensor structures motivated by first principles. Concerning scalar coefficients, mass corrections have been obtained by hypergeometric representations of Mellin–Barnes integrals. By our technique, the completeness of such kinematic singularity/zero-free tensor decomposition of the HLbL amplitude is explicitly checked.

Solutions of the sl2${\mathfrak {sl}_2}$qKZ equations modulo an integer

AbstractWe study the qKZ difference equations with values in the th tensor power of the vector representation , variables , and integer step . For any integer relatively prime to the step , we construct a family of polynomials in variables with values in such that the coordinates of these polynomials with respect to the standard basis of are polynomials with integer coefficients. We show that satisfy the qKZ equations modulo . Polynomials are modulo analogs of the hypergeometric solutions of the qKZ given in the form of multidimensional Barnes integrals.

Некоторые формулы дробного интегрирования от одной функции Фокса с пятью параметрами

The study obtained Riemann – Liouville and Erdelyi – Kober formulas for fractional integration of five-parameter special Fox function. The paper gives an integral representation of the function under consideration in terms of the Mellin – Barnes integral, conditions for absolute convergence, and asymptotic expansions both for large and small values of the argument. We obtained the formulas proved, using the above integral representation and known power rules for integration. For partial parameters, we got some well-known elementary and special functions.

Geometrical Methods for the Analytic Evaluation of Multiple Mellin–Barnes Integrals

Geometrical Methods for the Analytic Evaluation of Multiple Mellin–Barnes Integrals

Mellin–Barnes Integrals Related to the Lie Algebra u(N)

Mellin–Barnes Integrals Related to the Lie Algebra u(N)

To the Properties of One Fox Function

В работе рассматривается частный случай специальной функции Фокса с четырьмя параметрами, которая возникает в теории краевых задач для параболических уравнений c оператором Бесселя и дробной производной по времени. Целью исследования является получение некоторых рекуррентных соотношений, формул дифференцирования и интегрального преобразования рассматриваемой функции. При получении результатов работы в основном используется представление рассматриваемой функции через интеграл Меллина–Барнса. Также используются её асимптотические разложения при большом и малом значениях аргумента. С помощью указанного интегрального представления и некоторых известных формул для гамма-функции Эйлера, получены рекуррентные соотношения, связывающие функции с разными параметрами, а также функцию с её производной первого порядка. Получена формула дифференцирования n-го порядка. Исследуется несобственный интеграл первого рода, который содержит рассматриваемую функцию с двумя зависимыми параметрами из четырёх. Показывается, что этот несобственный интеграл может быть записан в терминах известной специальной функции Макдональда. При частных значениях параметров рассматриваемой в работе функции получаются некоторые известные элементарные и специальные функции. Результаты работы носят теоретический характер и будут полезны при исследовании краевых задач для вырождающихся параболических уравнений с производными дробного порядка по времени. The paper considers a particular case of a special Fox function with four parameters, which arises in the theory of boundary value problems for parabolic equations with a Bessel operator and a fractional time derivative. The research objective is to obtain some recurrence relations, formulas for differentiation and integral transformation of the function under consideration. The results are obtained through representation of the considered function in terms of the Mellin–Barnes integral. The function asymptotic expansions for large and small values of the argument are also used. Employing the integral representation and some wellknown formulas for the Euler gamma function, recurrent relations are obtained connecting functions with different parameters, as well as a function with its first-order derivative. A formula for differentiation of the nth order is obtained. The paper studies an improper integral of the first kind that includes the considered function with two dependent of the four parameters. We show that the improper integral can be written out in terms of the well-known special Macdonald function. With special values of the parameters of the considered function we obtain some well-known elementary and special functions. The results of the study are theoretical and applicable in the study of boundary value problems for degenerate parabolic equations with fractional time derivatives.

Nonterminating transformations and summations associated with some q-Mellin–Barnes integrals

In many cases one may encounter an integral which is of q-Mellin–Barnes type. These integrals are easily evaluated using theorems which have a long history dating back to Slater, Askey, Gasper, Rahman and others. We derive some interesting q-Mellin–Barnes integrals and using them we derive transformation and summation formulas for nonterminating basic hypergeometric functions. The cases which we treat include ratios of theta functions, the Askey–Wilson moments, nonterminating well-poised ϕ23, nonterminating very-well-poised W45, W78, products of two nonterminating ϕ12's, square of a nonterminating well-poised ϕ12, a nonterminating W910, two nonterminating W1112's and several nonterminating summations which arise from the Askey–Roy and Gasper integrals.

Wave Function for GL(n,ℝ) Hyperbolic Sutherland Model II. Dual Hamiltonians

Abstract Recently, we found Mellin–Barnes integrals, representing the wave function for $GL(n,{\mathbb {R}})$ hyperbolic Sutherland model. In present paper, we establish bispectral properties of this wave function with respect to dual Ruijesenaars–Macdonald operators.

Wave Function for GL(n, ℝ) Hyperbolic Sutherland Model

Abstract We obtain certain Mellin–Barnes integrals that present wave functions for $GL(n,{\mathbb {R}})$ hyperbolic Sutherland model with arbitrary positive coupling constant.

Некоторые формулы дробного интегрирования от одной функции Фокса с четырьмя параметрами

Solutions to many problems of mathematical physics, engineering and economics arc expressed through the so-called special functions. In the theory of special functions an important place is occupied by functions of the hypergeometric type. Many of them can be written in terms of the Meyer G-function. A generalization of the Meyer function is the Fox H-function. Some properties of this function can be obtained from its representation using the Mellin – Barnes integral. When deriving some formulas for this function for particular values of its parameters, due to the cumbersome writing of the Fox function, it is more convenient to use simplified notation. In this paper, we consider a special case of such a Fox function containing four parameters. For this function, Riemann-Liouville and Erdelyi-Kober fractional integration formulas arc obtained. An integral representation of the considered function h through the Mellin – Burns integral, we write out the conditions under which it converges absolutely, and the asymptotic expansions for this function for large and small values of the argument. The formulas proved in the paper are obtained using the indicated Mellin – Burns integral representation and the well-known integration formulas from power functions. For particular values of the parameters, the function under consideration yields some well-known elementary and special functions, and from the obtained formulas of fractional integration – the known integral values of these functions.

On the Laplace transform of the lognormal distribution: Analytic continuation and series approximations

We study the analytical properties of the Laplace transform of the lognormal distribution. Two integral expressions for the analytic continuation of the Laplace transform of the lognormal distribution are provided, one of which takes the form of a Mellin–Barnes integral. As a corollary, we obtain an integral expression for the characteristic function. We present two approximations for the Laplace transform of the lognormal distribution, both valid in ℂ∖(−∞,0]. In the last section, we discuss how one may use our results to compute the density of a sum of independent lognormal random variables.

Fundamental solution of a multi-dimensional distributed order fractional diffusion equation

In this paper, we consider the multi-dimensional distributed order fractional diffusion equation in the whole space, the half space and the corridor space. We use the Titchmarsh theorem to show that the fundamental solutions are expressed in terms of the Laplace-type integrals and can be applied for the Gauss–Laguerre quadrature. The mathematical identities and their relationships for the fundamental solutions in different dimensions are also discussed, and the alternative representation is given in terms of the Mellin–Barnes integrals for the fundamental solution in the whole space.

Orthogonal Polynomials with Ultra-Exponential Weight Functions: An Explicit Solution to the Ditkin–Prudnikov Problem

New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, we give an explicit solution to the Ditkin–Prudnikov problem (1966). The 3-term recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are derived. The method is based on differential properties of the involved special functions and their representations in terms of the Mellin–Barnes and Laplace integrals. A notion of the composition polynomial orthogonality is introduced. The corresponding advantages of this orthogonality to discover new sequences of polynomials and their relations to the corresponding multiple orthogonal polynomial ensembles are shown.

Coxeter group actions and limits of hypergeometric series

In this paper, we use combinatorial group theory and a limiting process to connect various types of hypergeometric series, and of relations among such series. We begin with a set S of 56 distinct translates of a certain function M, which takes the form of a Barnes integral, and is expressible as a sum of two very-well-poised $$_9F_8$$ hypergeometric series of unit argument. We consider a known, transitive action of the Coxeter group $$W(E_7)$$ on this set. We show that, by removing from $$W(E_7)$$ a particular generator, we obtain a subgroup that is isomorphic to $$W(D_6)$$ , and that acts intransitively on S, partitioning it into three orbits, of sizes 32, 12, and 12, respectively. Taking certain limits of the M functions in the first orbit yields a set of 32 J functions, each of which is a sum of two Saalschutzian $$_4F_3$$ hypergeometric series of unit argument. The original action of $$W(D_6)$$ on the M functions in this orbit is then seen to correspond to a known action of this group on this set of J functions. In a similar way, the image of each of the size-12 orbits, under a similar limiting process, is a set of 12 L functions that have been investigated in earlier works. In fact, these two image sets are the same. The limiting process is seen to preserve distance, except on pairs consisting of one M function from each size-12 orbit. Finally, each known three-term relation among the J and L functions is seen to be obtainable as a limit of a known three-term relation among the M functions.

Matrix model generating function for quantum weighted Hurwitz numbers

The KP -function of hypergeometric type serving as generating function for quantum weighted Hurwitz numbers is used to compute the Baker function and the corresponding adapted basis elements, expressed as absolutely convergent Laurent series in the spectral parameter. These are equivalently expressed as Mellin–Barnes integrals, analogously to Meijer G-functions, but with an infinite product of -functions as integral kernel. A matrix model representation is derived for the -function evaluated at trace invariants of an externally coupled matrix.

Some properties of the one-dimensional subordinated stable model

In this paper, we consider some properties of the model constructed from the one-dimensional stable processes by changing the time to a non-decreasing Lévy process. Our first result reveals a relation between this class of processes and the class of time-changed Brownian motions. Moreover, we describe the CGMY (Carr–Geman–Madan–Yor) model as subordinated stable process, and show the representation of the Lévy density of the corresponding subordinator via the Mellin–Barnes integral.

Analytical evaluation of certain integrals occurring in studies of wireless communications systems using the Mellin-transform method

This paper illustrates the effectiveness of the Mellin-transform (MT) method for the exact calculation of certain one-dimensional definite integrals occurring in studies of wireless communications systems. The MT method, though it often requires laborious calculations, is a straightforward and extremely powerful technique for the closed-form evaluation of definite integrals. Only basic information is provided with regard to the MT and the closely related class of Mellin–Barnes (MB) integrals. On this basis, the MT method is applied to several examples of integrals occurring in studies of fading channels and wireless transmission systems. Closed-form expressions are obtained, which are either new or provided without proof in the literature. The new formulas are suitable for numerical evaluation.

Asymptotics of the contour of the stationary phase and efficient evaluation of the Mellin-Barnes integral for the F3 structure function

A new approximation is proposed for the contour of the stationary phase of the Mellin--Barnes integrals in the case of its finite asymptotic behavior as ${\rm Re} z\to -\infty$. The efficiency of application of the proposed contour and the quadratic approximation to the contour of the stationary phase is compared by the example of the inverse Mellin transform for the structure function $F_3$. It is shown that, although for a small number of terms $N$ in quadrature formulas used to calculate integrals along these contours, the quadratic contour is more efficient, but for $N>20$ the asymptotic stationary phase integration contour gives better accuracy. The case of the $Q^2$-dependence of the $F_3$ structure function is also considered.

Hypercomplex Fock states for discrete electromagnetic Schrödinger operators: A Bayesian probability perspective

We present and study a new class of Fock states underlying to discrete electromagnetic Schrödinger operators from a multivector calculus perspective. This naturally lead to hypercomplex versions of Poisson–Charlier polynomials, Meixner polynomials, among other ones. The foundations of this work are based on the exploitation of the quantum probability formulation ‘à la Dirac’ to the setting of Bayesian probabilities, on which the Fock states arise as discrete quasi-probability distributions carrying a set of independent and identically distributed (i.i.d) random variables. By employing Mellin–Barnes integrals in the complex plane we obtain counterparts for the well-known multidimensional Poisson and hypergeometric distributions, as well as quasi-probability distributions that may take negative or complex values on the lattice hZn.