Euler Beta Research Articles

Analytic properties and numerical representations for constructing the extended beta function using logarithmic mean

<abstract><p>This paper aimed to obtain generalizations of both the logarithmic mean ($ \text{L}_{mean} $) and the Euler's beta function (EBF), which we call the extended logarithmic mean ($ \text{EL}_{mean} $) and the extended Euler's beta-logarithmic function (EEBLF), respectively. Also, we discussed various properties, including functional relations, inequalities, infinite sums, finite sums, integral formulas, and partial derivative representations, along with the Mellin transform for the EEBLF. Furthermore, we gave numerical comparisons between these generalizations and the previous studies using MATLAB R2018a in the form of tables and graphs. Additionally, we presented a new version of the beta distribution and acquired some of its characteristics as an application in statistics. The outcomes produced here are generic and can give known and novel results.</p></abstract>

Discrete phase space-continuous time relativistic Klein–Gordon and Dirac equations, and a new non-singular Yukawa potential

This paper deals with the second quantization of interacting relativistic Fermionic and Bosonic fields in the arena of discrete phase space and continuous time. The mathematical formulation involves partial difference equations. The corresponding Feynman diagrams and a new S^{#}-matrix theory is developed. In the special case of proton-proton Møller scattering via an exchange of a neutral meson, the explicit second order element langle f | S^{#}_{(2)} |i rangle is deduced. In the approximation of very low external three-momenta, a new Yukawa potential is explicitly derived from langle f | S^{#}_{(2)} |i rangle . Moreover, it is rigorously proved that this new Yukawa potential is divergence-free. The mass parameter of the exchanged meson may be set to zero to obtain a type of scalar Boson exchange between hypothetical Fermions. This provides a limiting case of a new Coulomb type potential directly from the new singularity free Yukawa potential. A divergence-free Coulomb potential between two Fermions at two discrete points is shown to be proportional to the Euler beta function. Within this relativistic discrete phase space continuous time, a single quanta is shown to occupy the hyper-tori S^{1}_{n^1} times S^{1}_{n^3} times S^{1}_{n^3} where S^{1}_{n} is a circle of radius sqrt{2n+1}.

A new Euler-Beta function model with statistical implementation related to the Mittag-Leffler-Kummer function

Recently, the realm related to Euler's Beta function has played a significant role in the development of special function theory. In this study, a new extension of the special function known as Euler's Beta function with respect to the Mittag-Leffler-Kummer function is introduced. Another formula of this new Beta function in terms of the Fox-Wright function is also presented. Numerous analytical properties of this new special function, such as several functional and summation relations, Mellin transforms, integral representations, and several derivative formulas, are studied. Furthermore, significant statistical implementations of Euler's Beta distribution are also discussed.

"On the generalized trapezoid and midpoint type inequalities involving Euler's beta function}"

"The main object of this paper is to present some generalizations of fractional integral inequalities involving Euler's beta function of Hermite-Hadamard type which cover the previously published result such as Riemann integral, Riemann-Liouville fractional integral, k-Riemann-Liouville fractional integral."

Integral transforms and probability distributions for a certain class of Fox–Wright type functions and its applications

Main focus of this paper is to introduce a new class of Fox–Wright type functions and obtain several new integral representations and various integral transforms such as Mellin transform, Whittaker transform, Jacobi transform, Gegenbauer transform, Laplace transform, Euler (Beta) transform, Pδ, Rν and Hankel transform. Moreover, a new probability density function is also introduced and several properties are studied. As application, fractional derivative and fractional integration formula are established. The results obtained in this paper, are presumably new and their importance is illustrated by several interesting consequences and examples. Furthermore, numerical computation and graphical representations have been expressed to validate the results.

Some properties of k-generalized Mittag Leffler function related to fractional calculus

This paper deals with the k-new generalized Mittag Leffler function. Some of its properties related to fractional calculus are presented viz. k-Weyl fractional integral and k-extended Euler beta integral transform, Whittaker integral transform. Some important special cases of the main results are also have been discussed.

Celestial operator product expansions and w1+\u221e symmetry for all spins

The operator product expansion of massless celestial primary operators of arbitrary spin is investigated. Poincaré symmetry is found to imply a set of recursion relations on the operator product expansion coefficients of the leading singular terms at tree-level in a holomorphic limit. The symmetry constraints are solved by an Euler beta function with arguments that depend simply on the right-moving conformal weights of the operators in the product. These symmetry-derived coefficients are shown not only to match precisely those arising from momentum-space tree-level collinear limits, but also to obey an infinite number of additional symmetry transformations that respect the algebra of w1+∞. In tree-level minimally-coupled gravitational theories, celestial currents are constructed from light transforms of conformally soft gravitons and found to generate the action of w1+∞ on arbitrary massless celestial primaries. Results include operator product expansion coefficients for fermions as well as those arising from higher-derivative non-minimal couplings of gluons and gravitons.

On a Unified Mittag-Leffler Function and Associated Fractional Integral Operator

The aim of this paper is to unify the extended Mittag-Leffler function and generalized Q function and define a unified Mittag-Leffler function. Both the extended Mittag-Leffler function and generalized Q function can be obtained from the unified Mittag-Leffler function. The Laplace, Euler beta, and Whittaker transformations are applied for this function, and generalized formulas are obtained. These formulas reproduce integral transformations of various deduced Mittag-Leffler functions and Q function. Also, the convergence of this unified Mittag-Leffler function is proved, and an associated fractional integral operator is constructed.

Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas

Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas

Celestial operator products of gluons and gravitons

The operator product expansion (OPE) on the celestial sphere of conformal primary gluons and gravitons is studied. Asymptotic symmetries imply recursion relations between products of operators whose conformal weights differ by half-integers. It is shown, for tree-level Einstein–Yang–Mills theory, that these recursion relations are so constraining that they completely fix the leading celestial OPE coefficients in terms of the Euler beta function. The poles in the beta functions are associated with conformally soft currents.

A Study of k-generalized Mittag-Leffler Type Function with Four Parameters

This study introduces a new generalization of the Mittag-Leffler function, the k-generalized Mittag-Leffler type function , , , ( ), γ δ α β E z k and investigates its interesting and important basic properties, such as useful relationships between Mittag-Leffler functions, recurrence relations, differential formulas, integral representations, and images of this function under the Euler (Beta), Laplace, and Whittaker transforms. Some interesting special cases of the main finding are also addressed and demonstrated to be related to some well-known ones.

Certain investigations in the field of special functions

Motivated mainly by a variety of applications of Euler's Beta, hypergeometric, and confluent hypergeometric functions together with their extensions in a wide range of research fields such asengineering, chemical, and physical problems. In this paper, we introduce modified forms of some extended special functions such as Gamma function, Beta function, hypergeometric function and confluent hypergeometric function by making use of the idea given in reference \cite{9}. Also, certain investigations including summation formulas, integral representations and Mellin transform of these modified functions are derived. Further, many known results are obtained asspecial cases of our main results.

Chemical Potential-Induced Wall State Transitions in Plant Cell Growth

The pH/T duality of acidic pH and temperature (T) action for the growth of grass shoots was examined in order to derive the phenomenological equation of wall properties for living plants. By considering non-meristematic growth as a dynamic series of state transitions (STs) in the extending primary wall, the critical exponents were identified, which exhibit a singular behaviour at a critical temperature, critical pH and critical chemical potential (μ) in the form of four power laws: f_{pi } left( tau right) propto left| tau right|^{beta - 1}, f_{tau } (pi ) propto left| pi right|^{1 - alpha }, g_{mu } (tau ) propto left| tau right|^{ - 2 - alpha + 2beta } and g_{tau } (mu ) propto left| mu right|^{2 - alpha }. The indices α and β are constants, while π and τ represent a reduced pH and reduced temperature, respectively. The convexity relation α + β ≥ 2 for practical pH-based analysis and β ≡ 2 “mean-field” value in microscopic (μ) representation were derived. In this scenario, the magnitude that is decisive is the chemical potential of the H+ ions, which force subsequent STs and growth. Furthermore, observation that the growth rate is generally proportional to the product of the Euler beta functions of T and pH, allowed to determine the hidden content of the Lockhart constant Ф. It turned out that the pH-dependent time evolution equation explains either the monotonic growth or periodic extension that is usually observed—like the one detected in pollen tubes—in a unified account.

Decomposition of Feynman integrals on the maximal cut by intersection numbers

We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss 2F1 hypergeometric function, and the Appell F1 function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to n-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.

A Study of the p-Adic Frobenius Lifts and p-Adic Periods, from a Deformation Theory Viewpoint

A canonical p-adic Frobenius lift is defined in the context of p-adic numbers, viewed as deformations of the corresponding finite field. Applications to p-adic periods are considered, including to the classical Euler gamma and beta functions and their p-adic analogues, from a cohomological point of view. Connections between various methods for computing scattering amplitudes are related to the moduli space problem and period domains.

Beta-type functions and the harmonic mean

For arbitrary f:left( a,infty right) rightarrow left( 0,infty right) ,age 0, the bivariable function B_{f}:left( a,infty right) ^{2}rightarrow left( 0,infty right) , related to the Euler Beta function, is considered. It is proved that B_{ftext { }}is a mean iff it is the harmonic mean H. Some applications to the theory of iterative functional equations are given.

Spectral zeta functions of graphs and the Riemann zeta function in the critical strip

We initiate the study of spectral zeta functions $\zeta_X$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $\zeta(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of $\zeta(s)$. We relate $\zeta_{\mathbb{Z}}$ to Euler's beta integral and show how to complete it giving the functional equation $\xi_{\mathbb{Z}}(1-s)=\xi_{\mathbb{Z}}(s)$. This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions $d$ we provide a meromorphic continuation of $\zeta_{\mathbb{Z}^d}(s)$ to the whole plane and identify the poles. From our aymptotics several known special values of $\zeta(s)$ are derived as well as its non-vanishing on the line $Re(s)=1$. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function $F_1$ via an Euler-type integral formula due to Picard.

A Mean Value Inequality for Euler's Beta Function

A Mean Value Inequality for Euler's Beta Function

New Integral Inequalities via Harmonically Convex Functions

In this paper, we establish some estimates, involving the Euler Beta function and the Hypergeometric function of the integral for the class of functions whose certain powers of the absolute value are harmonically convex.

Some applications of Retkes’ identity

We present some formulas for certain numeric sums related to the Riemann zeta function. The main tool used in our investigation is Retkes’ identity. We get a formula for \({\zeta{(3)}}\) with the Euler beta function in it.