Appell Function Research Articles

One-point thermal conformal blocks from four-point conformal integrals

We develop the thermal shadow formalism to study the conformal blocks decomposition in D-dimensional conformal field theory on Sβ1×SD−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{S}}_{\\beta}^1\ imes {\\mathbbm{S}}^{D-1} $$\\end{document}, where the temperature is T = β−1. It is demonstrated that both the 1-point thermal (T ≠ 0) conformal blocks and the 4-point plane (T = 0) conformal blocks are defined by the same 4-point conformal integral. It is shown that up to power prefactors the 1-point thermal conformal block is given by the fourth Appell function.

Bivariate Pareto–Feller Distribution Based on Appell Hypergeometric Function

The Pareto–Feller distribution has been widely used across various disciplines to model “heavy-tailed” phenomena, where extreme events such as high incomes or large losses are of interest. In this paper, we present a new bivariate distribution based on the Appell hypergeometric function with marginal Pareto–Feller distributions obtained from two independent gamma random variables. The proposed distribution has the beta prime marginal distributions as special case, which were obtained using a Kibble-type bivariate gamma distribution, and the stochastic representation was obtained by the quotient of a scale mixture of two gamma random variables. This result can be viewed as a generalization of the standard bivariate beta I (or inverted bivariate beta distribution). Moreover, the obtained bivariate density is based on two confluent hypergeometric functions. Then, we derive the probability distribution function, the cumulative distribution function, the moment-generating function, the characteristic function, the approximated differential entropy, and the approximated mutual information index. Based on numerical examples, the exact and approximated expressions are shown.

Extension of Delaunay normalisation for arbitrary powers of the radial distance

In the framework of perturbed Keplerian systems we deal with the Delaunay normalisation of a wide class of perturbations such that the radial distance is raised to an arbitrary real number γ. The averaged function is expressed in terms of the Gauss hypergeometric function 2F1 whereas the associated generating function is the so called Appell hypergeometric function F1. The Gauss hypergeometric function related to the average depends on the eccentricity, e, whereas the Appell function depends additionally on the eccentric anomaly, E, and both special functions are properly defined and evaluated for all e∈[0,1) and E∈[−π,π]. We analyse when the functions we determine can be extended to e=1. When the exponent of the radial distance is an integer, the usual values of the averaged and generating functions are recovered.

Apparent and emergent dark matter around a Schwarzschild black hole

Inspired by the two different dark matter frameworks that were studied recently: one that arises from the non-local effects of entanglement entropy as emergent gravity (characterized by the parameter ξ(M), and zero-point scale length l), and one from dark energy viewed as a superconducting medium (characterized by η(M), and screening length parameter λG), two black hole solutions spherically surrounded with these dark matter models were derived. The effect of these two frameworks on SMBH was analyzed through the resulting deviations in the null regions and the black hole shadow. In addition, constraints to the parameters ξ and η (at 3σ level) were found using the available EHT data for Sgr. A* and M87*. These constraints allow one to deduce the effective mass M, which causes uncertainties in the measurement. On the other hand, if the effective mass is known, one can also deduce the constants associated with ξ, and η. The former framework also introduces an Appell function, a hypergeometric function of two variables that separately allows the analysis of macroscopic and (hypothetical) microscopic black holes. This first framework was found to decrease the radii of the null regions respective to the Schwarzschild counterpart. The shadow radius, however, behaves reversibly. The result of the numerical analysis for the latter framework revealed increased values for the photonsphere and shadow radii. Remarkably, the study also showed that for SMBHs, the amplifying effects of λG are stronger than the scale length l. Finally, results of constraints, as an example, for the upper bound in ξ for M87* indicated that the effective mass causing the deviation was around 2.4×1020M⊙ given that the observed Milgrom’s constant is 5.4×10−10m/s2.

A family of algebraic curves and Appell series over finite fields

A family of algebraic curves and Appell series over finite fields

Celestial conformal blocks of massless scalars and analytic continuation of the Appell function F1

In celestial conformal field theory (CCFT), the 4d massless scalars are represented by 2d conformal operators with conformal dimensions h = h¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{h} $$\\end{document} = (1 + iλ)/2. The Mellin transform of 4d massless scalar amplitudes gives the conformal correlators of CCFT. We study the conformal block decomposition of these celestial correlators in CCFT and obtain the explicit blocks. This conformal block decomposition is highly nontrivial, even for the simplest 4d massless scalar amplitude. We use the analytic continuation of the Appell hypergeometric function F1 and the method of monodromy projection of conformal blocks, to achieve this block decomposition. This procedure is consistent with the crossing symmetry, in both the correlator-level and each explicit block-level. We also investigate its behavior in the conformal soft limit and find that the Appell hypergeometric function F1 does not reduce to the Gauss hypergeometric function. This is different from the block decomposition of celestial gluons we studied before, where the Appell hypergeometric function F1 reduces to the Gauss hypergeometric function. This difference comes from the shift of conformal dimensions and is the reason why we adopt the new method here for the block decomposition of celestial massless scalars.

EXPANSIONS OF KAMP ́E DE F ́ERIET HYPERGEOMETRIC FUNCTIONS

When studying the properties of the hypergeometric functions in two variables, expansion formulas are very important, allowing one to represent a function of two variables in the form of an infinite sum of products of several hypergeometric Gaussian functions, and this in turn facilitates the process of studying the properties of functions in two variables. Burchnall and Chaundy, in 1940--41, using the symbolic method, obtained more than 15 pairs of expansions for the second-order double hypergeometric Appell and Humbert functions. In order to find expansion formulas for functions depending on three or more variables, Hasanov and Srivastava introduced symbolic operators, with the help of which they were able to expand a whole class of hypergeometric functions of several variables. Hasanov, Turaev and Choi defined so-called H-operators that make it possible to find expansions for generalized hypergeometric functions of one variable. In addition, applications of these H-operators to the expansion of the hypergeometric functions of two and three variables of second order are known. On the other hand, thanks to the Kampe de Feriet functions, solutions of the boundary value problems for some degenerate and singular partial differential equations can be written in explicit forms. In this paper, expansion formulae are obtained for the hypergeometric Kampe de Feriet functions of the superior order. Some Kampe de Feriet functions are expanded in terms of the Appell and Humbert functions as illustrative examples

On Ramanujan's lost notebook and new tenth‐order like identities for second‐, sixth‐, and eighth‐order mock theta functions

AbstractRamanujan's lost notebook contains many mock theta functions and mock theta function identities not mentioned in his last letter to Hardy. For example, we find the four tenth‐order mock theta functions and their six identities. The six identities themselves are of a spectacular nature and were first proved by Choi. We also find eight sixth‐order mock theta functions in the lost notebook, but among their many identities there is only a single relationship like those of the tenth‐orders. Using Appell function properties of Hickerson and Mortenson, we discover and prove three new identities for the sixth‐order mock theta functions that are in the spirit of the six tenth‐order identities. We also include an additional nineteen tenth‐order like identities for various combinations of second‐, sixth‐, and eighth‐order mock theta functions.

On some new inequalities and fractional kinetic equations associated with extended gauss hypergeometric and confluent hypergeometric function

Fractional kinetic equations are of immense importance in describing and solving numerous intriguing problems of physics and astrophysics. Inequalities are important topics in special functions. In this paper, we studied the monotonicity of the extended Gauss and confluent hypergeometric function that are derived by using the inequalities on generalized beta function involving Appell series and Lauricella function. We also establish generalized fractional kinetic equation involving extended hypergeometric and confluent hypergeometric functions. The solutions of generalized fractional kinetic equation is derived and studied as an application of extended hypergeometric and confluent hypergeometric function using the General integral transform. The results obtained here are general and can be used to derive many new solutions of fractional kinetic equations involving various types of special functions.

A note on one-sided solutions for optimal stopping problems driven by Lévy processes

All optimal stopping problems with increasing, logconcave and right-continuous reward functions under Lévy processes have been shown to admit one-sided solutions. There is relatively little research attention, however, on constructing an explicit solution of the problem when the reward function is increasing and logconcave. Thus, this paper aims to explore the role of the monotonicity and logconcavity in the characterization of the optimal threshold value, particularly in terms of the ladder height process. In this paper, with additional smoothness assumptions on the reward function, we give an explicit expression for the optimal threshold value provided that the underlying Lévy process drifts to −∞ almost surely. In the particular case where the reward function is (x+)ν=(max{x,0})ν, ν∈(0,∞), we establish a connection indicating that the expression coincides with the positive root of the associated Appell function.

MULTIPLE EULER TYPE INTEGRAL REPRESENTATIONS FOR THE KAMP´E DE F´ERIET FUNCTIONS

By the aid of Appell, Humbert and Bessel functions, the integral representations for a Kamp´e de F´eriet function are found. The validity of integral representations for a Kamp´e de F´eriet function of general form are proved. Conditions, under which these representations are expressed in terms of products of two generalized hypergeometric functions are found. Examples, in which the integral representation of the Kamp´e de F´eriet function containing Appell, Humbert or Bessel functions, are identified.

On the two variables κ-Appell hypergeometric matrix functions

This work aims to introduce -Appell hypergeometric matrix functions and , where . New relations for these functions, such as the integral representation, important transformation formulas, infinite matrix summation and reduction formulas, are also presented. Finally, new results of the differential formulas associated with -Appell hypergeometric matrix functions are derived.

Completeness of trajectories associated to Appell hypergeometric functions

We examine the linear completeness of trajectories of eigenfunctions associated to non-linear eigenvalue problems, subject to Dirichlet boundary conditions on a segment. We pursue two specific goals. On the one hand, we establish that linear completeness persists for the non-linear Schrödinger equation, even when the trajectories lie far from those of the linear equation where bifurcations occur. On the other hand, we show that this is also the case for a fully non-linear version of this equation which is naturally associated with Appell hypergeometric functions. Both models shed new light on a framework for completeness in the non-linear setting, considered by L.E. Fraenkel over 40 years ago, that may have significant potential, but which does not seem to have received much attention.

Transformations for Appell series over finite fields and traces of Frobenius for elliptic curves

Using properties of certain character sums over finite fields, we here obtain some transformations for finite field Appell series F2⁎ and F4⁎ in terms of F34-Gaussian hypergeometric series. These motivate us to find a finite field analog for certain transformations satisfied by the classical Appell series F2 and F4. We also deduce an expression for the finite field Appell series F4⁎ in terms of F23-Gaussian hypergeometric series. Consequently, we investigate connections of trace of Frobenius endomorphism for certain families of elliptic curves and the finite field Appell series F4⁎.

Exact Closed-Form P fa Expressions for CA- and GO-CFAR Detectors in Gamma-Distributed Radar Clutter

In this correspondence, two exact closed-form expressions for the probability of false alarm have been derived assuming a homogeneous Gamma distributed clutter. These expressions concern both the cell averaging CFAR (CA-CFAR) and the greatest of CFAR (GO-CFAR) detectors. As it will be shown, the proposed expressions are given in terms of the Gauss hypergeometric function and the second Appell function. Our expressions are then examined and validated numerically, by comparing them to their counterparts computed using numerical integrals and Monte-Carlo simulations, considering various scenarios.

The incomplete matrix beta function and its application to first Appell hypergeometric matrix function

In this paper, we first introduce the incomplete beta matrix function B z ( M , K ) , z ∈ C , for parameter matrices M and K in C m × m using the neutrix limit and evaluation methods on their the partial derivatives. Also, we define a new extension of the Pochhammer matrix symbol in terms of the incomplete beta matrix functions. Furthermore, we then apply this extended matrix symbol for the incomplete first Appell hypergeometric matrix functions and obtain some of their properties.

Appell series over finite fields and modular forms

In this article, we express the special values of F4⁎, a finite field analogue of the classical Appell series F4, in terms of the Fourier coefficients of weight three modular forms. These modular forms are related to modular K3 surfaces. To obtain these results, we first prove the finite field analogues of classical identities connecting classical F23-hypergeometric series and F4-Appell series.

Some Properties of Incomplete First Appell Hypergeometric Matrix Functions

Some Properties of Incomplete First Appell Hypergeometric Matrix Functions

Fundamental solutions of a class of degenerate elliptic equation

In this paper, for the elliptic equation with two lines of different order of degeneration, given in the first quarter of the plane , four fundamental solutions are constructed that are expressed by Appell's hypergeometric functions of two variables. By means the decomposition formula of the Appell's function in products of the Gauss hypergeometric function, it is proved that the constructed fundamental solutions possess a logarithmic singularity for .

A Note on Certain General Transformation Formulas for the Appell and the Horn Functions

In a number of problems in applied mathematics, physics (theoretical and mathematical), statistics, and other fields the hypergeometric functions of one and several variables naturally appear. Hypergeometric functions in one and several variables have several known applications today. The Appell’s four functions and the Horn’s functions have shown to be particularly useful in providing solutions to a variety of problems in both pure and applied mathematics. The Hubbell rectangular source and its generalization, non-relativistic theory, and the hydrogen dipole matrix elements are only a few examples of the numerous scientific and chemical domains where Appell functions are used. The Appell series is also used in quantum field theory, especially in the evaluation of Feynman integrals. Additionally, since 1985, computational sciences such as artificial intelligence (AI) and information processing (IP) have used the well-known Horn functions as a key idea. In literature, there have been published a significant number of results of double series in particular of Appell and Horn functions in a series of interesting and useful research publications. We find three general transformation formulas between Appell functions F2 and F4 and two general transformation formulas between Appell function F2 and Horn function H4 in the present study, which are mostly inspired by their work and naturally exhibit symmetry. By using the generalizations of the Kummer second theorem in the integral representation of the Appell function F2, this is accomplished. As special cases of our main findings, both previously known and new results have been found.