Mellin-Barnes Integral Representation Research Articles

Bessel functions on GL(n), I

In the context of the Kuznetsov trace formula, we outline the theory of the Bessel functions on GL(n) as a series of conjectures designed as a blueprint for the construction of Kuznetsov-type formulas with given ramification at infinity. We are able to prove one of the conjectures at full generality on GL(n) and most of the conjectures in the particular case of the long Weyl element; as with previous papers, we give some unconditional results on Archimedean Whittaker functions, now on GL(n) with arbitrary weight. We expect the heuristics here to apply at the level of real reductive groups. A forthcoming paper will address the initial conjectures up to Mellin-Barnes integral representations in the case of GL(4) Bessel functions.

An analysis of the Yang 𝕐-function class extension through its incomplete functions

The new generalized hypergeometric function (Yang [Formula: see text]-function) described by the contour-type Mellin–Barnes integral representation serves as the inspiration for this study. The incomplete Yang [Formula: see text]-functions [Formula: see text] and [Formula: see text] that we shall introduce here are the appropriate extension of a class of [Formula: see text]-functions by merit of the gamma functions of incomplete type, [Formula: see text] and [Formula: see text]. In this work, our objective is to study several remarkable characteristics of these newly introduced incomplete [Formula: see text]-functions viz; reduction formula, derivative, various integral transformations. Additionally, using these newly established incomplete functions, we formulate and solve a generalized fractional kinetic equation in terms of these incomplete [Formula: see text]-functions. Further, we also propose the incomplete [Formula: see text]-functions [Formula: see text] and [Formula: see text] corresponding to a new generalization of the [Formula: see text]-function.

Lerch $\Phi$ Asymptotics

We use a Mellin-Barnes integral representation for the Lerch transcendent $\Phi(z,s,a)$ to obtain large $z$ asymptotic approximations. The simplest divergent asymptotic approximation terminates in the case that $s$ is an integer. For non-integer $s$ the asymptotic approximations consists of the sum of two series. The first one is in powers of $(\ln z)^{-1}$ and the second one is in powers of $z^{-1}$. Although the second series converges, it is completely hidden in the divergent tail of the first series. We use resummation and optimal truncation to make the second series visible.

On the solution of multi-term time fractional diffusion-wave equation involving ultra-hyperbolic operator

A diffusion-wave equation with multi-term Hilfer fractional derivatives (HFDs) in time and ultra-hyperbolic operator (UHO) in space has been considered. Fundamental solution of the fractional diffusion-wave equation is obtained by using Laplace and Fourier transform with Mellin-Barnes integral representation. The solution obtained involved the Fox H-function. In addition, we provide some special cases of diffusion-wave equation.

Borel (α,β)-multitransforms and quantum Leray–Hirsch: Integral representations of solutions of quantum differential equations for [formula omitted]-bundles

In this paper, we address the integration problem of the isomonodromic system of quantum differential equations (qDEs) associated with the quantum cohomology of P1-bundles on Fano varieties. It is shown that bases of solutions of the qDE associated with the total space of the P1-bundle can be reconstructed from the datum of bases of solutions of the qDE associated with the base space. This represents a quantum analog of the classical Leray–Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in [17], called Borel(α,β)-multitransforms. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer–Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen P1-bundle. When applied to projective bundles on products of projective spaces, our results give Mellin–Barnes integral representations of solutions of qDEs. As an example, we show how to integrate the qDE of blow-up of P2 at one point via Borel multitransforms of solutions of the qDE of P1.

Specializations of partial differential equations for Feynman integrals

Starting from the Mellin–Barnes integral representation of a Feynman integral depending on a set of kinematic variables zi, we derive a system of partial differential equations w.r.t. new variables xj, which parameterize the differentiable constraints zi=yi(xj). In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.

Some properties relating to the Mittag–Leffler function of two variables

An attempt is made here to study the Mittag–Leffler function with two variables. Its various properties including integral and operational relationships with other known Mittag–Leffler functions of one variable, pure and differential recurrence relations, Euler transform, Laplace transform, Mellin transform, Whittaker transform, Mellin–Barnes integral representation, and its relationship with Wright hypergeometric function are investigated and established. Also, properties of the Mittag–Leffler function of two variables associated with fractional calculus operators are considered.

Residue Sum Formula for Pricing Options under the Variance Gamma Model

We present and prove a triple sum series formula for the European call option price in a market model where the underlying asset price is driven by a Variance Gamma process. In order to obtain this formula, we present some concepts and properties of multidimensional complex analysis, with particular emphasis on the multidimensional Jordan Lemma and the application of residue calculus to a Mellin–Barnes integral representation in C3, for the call option price. Moreover, we derive triple sum series formulas for some of the Greeks associated to the call option and we discuss the numerical accuracy and convergence of the main pricing formula.

Time-Changed Dirac–Fokker–Planck Equations on the Lattice

A time-changed discretization for the Dirac equation is proposed. More precisely, we consider a Dirac equation with discrete space and continuous time perturbed by a time-dependent diffusion term $\sigma^2Ht^{2H-1}$ that seamlessly describes a latticizing version of the time-changed Fokker-Planck equation carrying the Hurst parameter $0<H<1$. Our model problem formulated on the space-time lattice $\mathbb{R}_{h,\alpha}^n\times [0,\infty)$ ($h>0$ and $0<\alpha<\frac{1}{2}$) preserves the main features of the Dirac-K\"ahler type discretization over the space-time lattice $h\mathbb{Z}^n\times [0,\infty)$ in case of $\alpha,H \rightarrow 0$, and encompasses a regularization of Wilson's approach [Physical review D, 10(8), 2445, 1974] for values of $H$ in the range $0<H\leq \frac{1}{2}$ (limit condition $\alpha \rightarrow \frac{1}{2}$). The main focus here is the representation of the solutions by means of discrete convolution formulae involving a kernel function encoded by (unnormalized) Hartman-Watson distributions -- ubiquitous on stochastic processes of Bessel type -- and the solutions of a semi-discrete equation of Klein-Gordon type. Namely, on our main construction the ansatz function $\widehat{\varPsi}_H(y)$ appearing on the discrete convolution representation may be rewritten as a Mellin convolution type integral involving the solutions $\varPsi(x,t|p)$ of a semi-discrete equation of Klein-Gordon type and a L\'evy one-sided distribution $L_H(u)$ in disguise. Interesting enough, by employing Mellin-Barnes integral representations it turns out that the underlying solutions of Klein-Gordon type may be represented through generalized Wright functions of type ${~}_1\Psi_1$, that converge uniformly in case that the quantity $\alpha+\frac{1}{2}$ may be regarded as an lower estimate for the Hurst parameter in the superdiffusive case (that is, if $\alpha+\frac{1}{2}\leq H<1$).

A Natural Representation of Functions that Facilitates `Exact Learning'

We present a collection of mathematical tools and emphasise a fundamental representation of analytic functions. Connecting these concepts leads to a framework for `exact learning', where an unknown numeric distribution could in principle be assigned an exact mathematical description. This is a new perspective on machine learning with potential applications in all domains of the mathematical sciences and the generalised representations presented here have not yet been widely considered in the context ofmachine learning and data analysis. The moments of a multivariate function or distribution are extracted using a Mellin transform and the generalised form of the coefficients is trained assuming a highly generalised Mellin-Barnes integral representation. The fit functions use many fewer parameters contemporary machine learning methods and any implementation that connects these concepts successfully will likely carry across to non-exact problems and provide approximate solutions. We compare the equations for theexact learning method with those for a neural network which leads to a new perspective on understanding what a neural network may be learning and how to interpret the parameters of those networks.

ON MELLIN-BARNES INTEGRAL REPRESENTATIONS FOR GKZ HYPERGEOMETRIC FUNCTIONS

We consider Mellin-Barnes integral representations of GKZ hypergeometric equations. We construct integration contours in an explicit way and show that suitable analytic continuations give rise to a basis of solutions.

Analytic Continuation for Solutions to the System of Trinomial Algebraic Equations

In the paper, we deal with the problem of getting analytic continuations for the monomial function associated with a solution to the reduced trinomial algebraic system. In particular, we develop the idea of applying the Mellin-Barnes integral representation of the monomial function for solving the extension problem and demonstrate how to achieve the same result following the fact that the solution to the universal trinomial system is polyhomogeneous. As a main result, we construct Puiseux expansions (centred at the origin) representing analytic continuations of the monomial function

Eigen spectra and wave functions of the massless Dirac fermions under the nonuniform magnetic fields in graphene

In this research, firstly, by using the new form of Dirac-Weyl equation and the series method with submitting more suitable details, the energy spectrum and wave functions of the massless Dirac fermions are calculated under the inhomogeneous and q-deformed spatially magnetic fields. Although, we discussed about the results of the energy levels, further, we obtained the wave function as the Hessenberg determinant with calculating the elements of it as exact. On the other hand, by using the Mellin-Barnes integral representation and Hurwitz zeta function, we have achieved the thermodynamic physical quantities of the Dirac-Weyl fermions in the absence of a magnetic field for inside of the graphene quantum dot. Finally, our numerical results for the wave functions and probability densities are presented too.

Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process

In this paper, the one-dimensional α-fractional diffusion equation is revisited. This equation is a particular case of the time- and space-fractional diffusion equation with the quotient of the orders of the time- and space-fractional derivatives equal to one-half. First, some integral representations of its fundamental solution including the Mellin-Barnes integral representation are derived. Then a series representation and asymptotics of the fundamental solution are discussed. The fundamental solution is interpreted as a probability density function and its entropy in the Shannon sense is calculated. The entropy production rate of the stochastic process governed by the α-fractional diffusion equation is shown to be equal to one of the conventional diffusion equation.

Spectral zeta function and non-perturbative effects in ABJM Fermi-gas

The exact partition function in ABJM theory on three-sphere can be regarded as a canonical partition function of a non-interacting Fermi-gas with an unconventional Hamiltonian. All the information on the partition function is encoded in the discrete spectrum of this Hamiltonian. We explain how (quantum mechanical) non-perturbative corrections in the Fermi-gas system appear from a spectral consideration. Basic tools in our analysis are a Mellin-Barnes type integral representation and a spectral zeta function. From a consistency with known results, we conjecture that the spectral zeta function in the ABJM Fermi-gas has an infinite number of "non-perturbative" poles, which are invisible in the semi-classical expansion of the Planck constant. We observe that these poles indeed appear after summing up perturbative corrections. As a consequence, the perturbative resummation of the spectral zeta function causes non-perturbative corrections to the grand canonical partition function. We also present another example associated with a spectral problem in topological string theory. A conjectured non-perturbative free energy on the resolved conifold is successfully reproduced in this framework.

English

The present paper is the investigation of certain properties of generalized Bessel-Maitland function, written in the form ࡶ ࣇ ,ࢗ ࣆ ,ࢽ (ࢠ ) = ∑ � ஶ ࢔ୀ૙ (ࢽ )ࢗ࢔ (ࢠ )࢔ ࢔ !ડ (ࣆ࢔ ାࣇା૚ ) , where ࣆ ,ࣇ ,ࢽ∈ ࡯ ;ࡾࢋ (ࣆ ) ≥ ૙ , ࡾࢋ (ࣇ ) ≥ −૚ ,ࡾࢋ (ࢽ ) ≥ ૙ and ࢗ∈ (૙ ,૚ )∪ ࡺ. For the function ࡶ ࣇ ,ࢗ ࣆ ,ࢽ (ࢠ ) , a number of results including differentiation and integration formulas, Mellin-Barnes integral representation, Laplace transform, Euler transform, ࢑ -transform, Varma transform, Mellin transform. Various relationship with other functions including Fox's ࡴ-function and Wright hypergeometric function were also established. In the end certain relations have been obtained by using the Riemann-Liouville fractional integrals and derivatives.

Multi-parametric mittag-leffler functions and their extension

Abstract This paper surveys one of the last contributions by the late Professor Anatoly Kilbas (1948–2010) and research made under his advisorship. We briefly describe the historical development of the theory of the discussed multi-parametric Mittag-Leffler functions as a class of the Wright generalized hypergeometric functions. The method of the Mellin-Barnes integral representations allows us to extend the considered functions to the case of arbitrary values of parameters. Thus, the extended Mittag-Leffler-type functions appear. The properties of these special functions and their relations to the fractional calculus are considered. Our results are based mainly on the properties of the Fox H-functions, as one of the widest class of special functions.

On Parabolic Whittaker Functions

We derive a Mellin-Barnes integral representation for solution to generalized (parabolic) quantum Toda lattice introduced in \cite{GLO}, which presumably describes the $(S^1\times U_N)$-equivariant Gromov-Witten invariants of Grassmann variety.

Baxter Operator and Archimedean Hecke Algebra

In this paper we introduce Baxter integral \({\mathcal{Q}}\) -operators for finite-dimensional Lie algebras \({\mathfrak{gl}_{\ell+1}}\) and \({\mathfrak{so}_{2\ell+1}}\) . Whittaker functions corresponding to these algebras are eigenfunctions of the \({\mathcal{Q}}\)-operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G = GL(ℓ + 1) proved earlier by Stade. We also identify eigenvalues of the Baxter \({\mathcal{Q}}\)-operator acting on Whittaker functions with local Archimedean L-factors. The Baxter \({\mathcal{Q}}\)-operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra \({\mathcal {H}(G(\mathbb{R}), K)}\) , K being a maximal compact subgroup of G. Finally we stress an analogy between \({\mathcal{Q}}\)-operators and certain elements of the non-Archimedean Hecke algebra \({\mathcal {H}(G(\mathbb{Q}_p),G(\mathbb{Z}_p))}\) .

On a generalization of Mittag-Leffler function and its properties

Let s and z be complex variables, Γ ( s ) the Gamma function, and ( s ) ν = Γ ( s + ν ) Γ ( s ) for any complex ν the generalized Pochhammer symbol. The principal aim of the paper is to investigate the function E α , β γ , q ( z ) = ∑ n = 0 ∞ ( γ ) q n Γ ( α n + β ) z n n ! , where α , β , γ ∈ C ; Re ( α ) > 0 , Re ( β ) > 0 , Re ( γ ) > 0 and q ∈ ( 0 , 1 ) ∪ N . This is a generalization of the exponential function exp ( z ) , the confluent hypergeometric function Φ ( γ , α ; z ) , the Mittag-Leffler function E α ( z ) , the Wiman's function E α , β ( z ) and the function E α , β γ ( z ) defined by Prabhakar. For the function E α , β γ , q ( z ) its various properties including usual differentiation and integration, Laplace transforms, Euler (Beta) transforms, Mellin transforms, Whittaker transforms, generalised hypergeometric series form, Mellin–Barnes integral representation with their several special cases are obtained and its relationship with Laguerre polynomials, Fox H-function and Wright hypergeometric function is also established.