Gamma Function Research Articles

Correction to: Evaluation of incomplete gamma functions using downward recursion and analytical relations

Correction to: Evaluation of incomplete gamma functions using downward recursion and analytical relations

Dirichlet Integral and Its Generalization by Different Methods

Dirichlet integral, as the integration of sine function over x from positive infinity to 0, or negative infinity, is known as one of the most important concepts in mathematical analysis. In probability and statistics, Dirichlet integral is used in calculating expectation and variance. In physics, Dirichlet integral is used in describing the movement of a charged particle in specific electromagnetic fields, and the wave function of particles in quantum physics. In computer science, Dirichlet integral is used in the optimization of neural network. In economics and management, Dirichlet integral is used in the optimization of production and distribution, the estimation of the risk and return in investment. In this paper, Dirichlet Integral and its generalization are studied. This paper provides several methods including residue theorem and Feynman’s trick to determine the value of Dirichlet integral. Meanwhile, this paper extends Dirichlet integral to the case of n -th power and deduces a general solution by gamma function for the integral.

Economic Load Dispatch Problem Analysis Based on Modified Moth Flame Optimizer (MMFO) Considering Emission and Wind Power

In electrical power system engineering, the economic load dispatch (ELD) problem is a critical issue for fuel cost minimization. This ELD problem is often characterized by non-convexity and subject to multiple constraints. These constraints include valve-point loading effects (VPLEs), generator limits, emissions, and wind power integration. In this study, both emission constraints and wind power are incorporated into the ELD problem formulation, with the influence of wind power quantified using the incomplete gamma function (IGF). This study proposes a novel metaheuristic algorithm, the modified moth flame optimization (MMFO), which improves the traditional moth flame optimization (MFO) algorithm through an innovative flame selection process and adaptive adjustment of the spiral length. MMFO is a population-based technique that leverages the intelligent behavior of flames to effectively search for the global optimum, making it particularly suited for solving the ELD problem. To demonstrate the efficacy of MMFO in addressing the ELD problem, the algorithm is applied to four well-known test systems. Results show that MMFO outperforms other methods in terms of solution quality, speed, minimum fuel cost, and convergence rate. Furthermore, statistical analysis validates the reliability, robustness, and consistency of the proposed optimizer, as evidenced by the consistently low fitness values across iterations.

The Spectral Gamma Function of gas absorption coefficient and its application to analysis and enhancement of the correlated spectral modeling of radiative transfer in non-uniform gaseous media

A novel Spectral Gamma Function is introduced to address the limitations associated with the “correlated” spectrum assumption in radiative transfer modeling in non-uniform gaseous media. The Spectral Gamma Function combines high-resolution gas absorption spectra with their distribution functions, capturing the “correlated” properties of absorption spectra at different thermodynamic states. This concept is applicable both to narrow bands and the full spectrum, enabling the construction of truly “correlated” models and the calculation of statistical relationships between absorption spectra. The paper presents the definition and properties of the Spectral Gamma Function. The Function 1) allows the construction of sets of truly “correlated” high-resolution spectra from which the “distance” to a real set of gas spectra can be characterized; 2) provides a simple way to evaluate Spearman's rank correlation between spectra in distinct states yielding useful information about the statistical relationship between these spectra; and 3) allows for the approximation of a real set of Line-by-Line (LBL) data as a non-linear combination of “correlated” models called MetaLBL modeling. Additionally, the function may be used to demonstrate the optimality of the Rank Correlated Spectral Line Weighted-sum-of-gray-gases (RC-SLW) Model in the space of “correlated” models, thus demonstrating that the RC-SLW Model represents the limit of accuracy in the family of correlated models. This work represents the first proposal of such a development extending modeling approaches beyond those inherent in the assumption of correlated spectra in the field of gas radiation modeling, offering promising insights for practical applications. Moreover, the MetaLBL modeling strategy is applied in a test case from atmospheric sciences for which models with a user-specified accuracy are constructed.

Microphysical characteristics of torrential predecessor rain events over the Yangtze River Delta Area and the related tropical cyclones

The precipitation structures and microphysical characteristics of predecessor rain events (PREs) over the Yangtze River Delta area and related tropical cyclones (TCs) from 2014 to 2019 were investigated using Dual-frequency Precipitation radar data from Global Precipitation Measurement (GPM) for drop size distributions (DSDs). Results showed that the total mean rain rate of PREs was larger than that of TCs, primarily due to higher convective and stratiform rain rates, with enhanced fractional coverage of convective rain in PREs. Examination of microphysical characteristics revealed that a greater quantity of small-sized droplets and a smaller quantity of medium- as well as large-sized droplets contributed towards PREs in comparison with TCs. The conclusion still holds when partitioning DSDs based on different precipitation rate categories. Further investigation of DSDs using gamma functions illustrated that precipitation in PREs had lower average mass-weighted diameters (Dm) and enhanced normalized number concentration (Nw) compared with TCs; partitioning precipitation into convective and stratiform components illustrated a larger Dm and lower Nw in TCs than PREs. The analysis of microphysical and thermodynamical processes using the reanalysis data indicates that relatively intense convective activity with drier conditions may be favorable to enhancing raindrop growth through collision-coalescence processes, as a result of larger Dm in TCs than PREs. The empirical relations (Z–R algorithms) applied in different rain regimes (stratiform, convective, and total PREs) revealed significant diversities, relying on weather conditions and geographical locations. Plain language summaryThe Yangtze River Delta area is an important economic belt in China. Under climate change, observed frequent occurrences of weather extremes of heavy rainfall and tropical cyclones (TCs) exert adverse effects on economic development in this region. Thus, a deep understanding of the mechanism of TCs torrential rainfall in the Yangtze River Delta area is urgently necessary. In this study, we investigated the precipitation patterns and microphysical characteristics of predecessor rain events (PREs) in the Yangtze River Delta region, and their association with TCs in the South China Sea-Western North Pacific Ocean (SCS-WNPO) area from 2014 to 2019. We found that PREs had a higher total mean precipitation rate than TCs. Further examination using gamma functions demonstrated that PREs exhibited lower average mass-weighted diameters (Dm) and higher normalized intercept parameters (Nw) than TCs. This pattern persisted when distinguishing between convective and stratiform precipitation components. We believe that our study makes a significant contribution to the literature because these results provide valuable insights into the distinct precipitation characteristics of PREs and TCs in the study region and contribute to a better understanding of tropical cyclone-related rainfall patterns, and act as a scientific basis for disaster mitigation.

Starlikeness and Convexity of Generalized Bessel-Maitland Function

The main objective of this research is to examine a specific sufficiency criteria for the starlikeness and convexity of order δ, k-uniform starlikeness, k-uniform convexity, lemniscate starlikeness and convexity, exponential starlikeness and convexity, uniform convexity of the Generalized Bessel-Maitland function. Applications of these conclusions to the concept of corollaries are also provided. Additionally, an illustrated representation of these outcomes will be presented. So functional inequalities involving gamma function will be the main research tools of this exploration. The outcomes from this study generalize a number of conclusions from earlier studies.

Properties and Applications of Complex Fractal–Fractional Operators in the Open Unit Disk

A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function: Γℓ(.)). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows the normalization associated with every operator. We investigate a number of significant geometric features thanks to this. Additionally, some integrals, such the Alexander and Libra integral operators, are associated with these operators. Simple power functions are among the illustrations that are provided. Additionally, the formulation of the discrete ℓ−fractal–fractional operators is conducted. We demonstrate that well-known examples are involved in the extended operators.

Extension of Pochhammer symbol, generalized hypergeometric function and τ-Gauss hypergeometric function

Abstract We introduce new extension of the extended Pochhammer symbol and gamma function by using the extended Mittag-Leffler function. We also present extension of the generalized hypergeometric function as well as some of their special cases by using this extended Pochhammer symbol. Further, we define the extension of the τ-Gauss hypergeometric function. Integral and derivative formulas involving the Mellin transform and fractional calculus techniques associated with this extended τ-Gauss hypergeometric function are also given. Also, new extended τ-Gauss hypergeometric function also provides a few more interesting and well-known results. This enriches the theory of special functions. The obtained results are believed to be newly presented.

Composition of some positive linear integral operators

Abstract This article is devoted to constructing sequences of integral operators with the same Voronovskaja formula as the generalized Baskakov operators, but having different behavior in other respects. For them, we investigate the eigenstructure, the inverses, and the corresponding Voronovskaja type formulas. A general result of Voronovskaja type for composition of operators is given and applied to the new operators. The asymptotic behavior of differences between the operators is investigated, and as an application, we obtain a formula involving Euler’s gamma function.

High dynamic range image tone mapping based on variational image decomposition and color correction

Tone mapping aims to reproduce high dynamic range (HDR) images on low dynamic range (LDR) display devices. The layer-decomposition-based tone mapping algorithms usually decompose HDR images into base and detail layers and process them accordingly. Since the detail layer is not refined again, these algorithms may suffer from halo artifacts, relatively dull color, and over-enhancement problems. To solve these problems, we introduce the variational image decomposition model and color correction into the field of tone mapping, and propose a tone mapping algorithm that combines the three-part variational image decomposition model with a color correction scheme based on minimum color loss. In contrast to traditional layer-decomposition-based tone mapping algorithms, we employ the three-part decomposition model that decomposes the HDR image into three components: a cartoon component, a texture component, and a high-frequency detail component. The cartoon component is compressed using a gamma function, while the texture and high-frequency detail components are each enhanced using the nonlinear stretching functions. This approach allows for better detail enhancement while preserving the smoothness of the image. In addition, we further use the color correction scheme based on minimum color loss to solve the color problem caused by the layer-decomposition-based tone mapping algorithms. Experimental results show that our proposed tone mapping algorithm can reduce the halo artifacts and preserve the edge details while outperforming some state-of-the-art tone mapping algorithms.

Inequalities for Basic Special Functions Using Hölder Inequality

Let p,q≥1 be two real numbers such that 1p+1q=1, and let a,b∈R be two parameters defined on the domain of a function, for example, f. Based on the well known Hölder inequality, we propose a generic inequality of the form |f(ap+bq)|≤|f(a)|1p|f(b)|1q, and show that many basic special functions, such as the gamma and polygamma functions, Riemann zeta function, beta function and Gauss and confluent hypergeometric functions, satisfy this type of inequality. In this sense, we also present some particular inequalities for the Gauss and confluent hypergeometric functions to confirm the main obtained inequalities.

Reciprocal Hyperbolic Series of Ramanujan Type

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z=F12(1/2,1/2;1;x) and z′=dz/dx. When a certain parameter in these series is equal to π, the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

On the Containment of the Unit Disc Image by Analytical Functions in the Lemniscate and Nephroid Domains

Suppose that A1 is a class of analytic functions f:D={z∈C:|z|<1}→C with normalization f(0)=1. Consider two functions Pl(z)=1+z and ΦNe(z)=1+z−z3/3, which map the boundary of D to a cusp of lemniscate and to a twi-cusped kidney-shaped nephroid curve in the right half plane, respectively. In this article, we aim to construct functions f∈A0 for which (i) f(D)⊂Pl(D)∩ΦNe(D) (ii) f(D)⊂Pl(D), but f(D)⊄ΦNe(D) (iii) f(D)⊂ΦNe(D), but f(D)⊄Pl(D). We validate the results graphically and analytically. To prove the results analytically, we use the concept of subordination. In this process, we establish the connection lemniscate (and nephroid) domain and functions, including gα(z):=1+αz2, |α|≤1, the polynomial gα,β(z):=1+αz+βz3, α,β∈R, as well as Lerch’s transcendent function, Incomplete gamma function, Bessel and Modified Bessel functions, and confluent and generalized hypergeometric functions.

Model based fitting of pattern reversal visually evoked potentials provides a reliable characterization of waveform components

ObjectiveTo introduce a novel approach to analyzing pattern reversal visual evoked potentials (prVEPs) using a difference-of-gammas model-based fitting method. MethodsprVEP was recorded from uninjured youth ages 11–19 years during pre- or post-season sports evaluation. A difference-of-gammas model fit was used to extract the amplitude, peak time, and peak width of each of four gamma components. The within session reliability and stability of fits across a 6-month period were determined. To demonstrate an application of this analysis, changes in parameters across age were determined. ResultsA difference-of-gammas model consisting of four gamma functions was fit to the prVEP of 151 youth. Peak times and amplitudes of functions corresponded to standard measures of the N75, P100, and N135 components respectively, and a late gamma peak (mean peak time 219 ms). We extracted the peak width, which increased with each subsequent temporal peak. Parameter fits were reliable within sessions (correlation coefficient > 0.92 for all measured parameters; good agreement on Bland-Altman calculation) and were stable between sessions separated by less than 6 months (correlation coefficient > 0.90). Standard peak analysis metrics extracted from the difference-of-gamma model fits were largely consistent with gold-standard peak analysis measurements. ConclusionsThe difference-of-gammas model provides a stable and reliable within-participant representation of the global temporal variability of prVEP waveforms across a large sample of youth. SignificanceUsing difference-of-gammas model to characterize the global temporal variability of the prVEP waveform offers a promising direction to enhance analysis for identifying and following subtle changes in neurologic conditions.

On Positive Linear Operators linking Gamma, Mittag-Leffler and Wright Functions

On Positive Linear Operators linking Gamma, Mittag-Leffler and Wright Functions

Consistent solutions for simultaneous dynamic water, solute, stable isotope and radon balances to estimate average exchange fluxes between surface water bodies and groundwater

Many papers show that water, conservative solute, stable isotope and radon balances can be applied to the study of surface water – groundwater interaction. When undertaken simultaneously, such analysis provides an even more powerful method for estimating groundwater inflows and outflows, i.e. the exchange fluxes that for most people are hardest to estimate. The overarching objective of this paper is to provide practitioners with a consistent conceptual and simulation modelling framework with which to achieve a rapid initial understanding of surface water – groundwater interaction, as well as to guide the efficient and targeted acquisition of field data. This paper makes several important contributions. First, after an introduction that explains the complexities of groundwater flow patterns near surface water bodies, a set of general balance equations is developed using consistent terminology and notation. Second, after focusing on a simple conceptual model with steady groundwater inflow I, groundwater outflow O, precipitation P and ever-present evaporation E, analytical solutions are developed for transient and steady state conditions: one water balance solution is found to apply in all situations; solutions for the solute balance equation are developed for 23 different subcases; a transient balance equation for stable isotopes written in terms of 1+δ is found to apply in all situations; and a new transient solution for radon is found in terms of the upper incomplete gamma function. On the path towards developing the isotope balance equation, we present a rigorous approach, based on coupled nonlinear balance equations for abundant and rare isotope pairs (1H and 2H, then 16O and 18O), using isotope ratios and the fractions of each isotope. Third, we describe a Lake Water Balance Calculator (LWBC) written using GoldSim simulation software, a tool to help others to conduct the simultaneous analyses that we promote; we use GoldSim’s numerical methods to solve the general balance equations, we implement the analytical solutions to demonstrate that both GoldSim’s numerical methods and the analytical solutions are correct, and we also show that the 1+δ approximation is equivalent to the rigorous coupled equations, at least for heavy stable isotopes with very low abundance. Fourth, we describe field studies: we provide three examples of applications of LWBC to short-term studies of Perry Lakes East, Lake Jasper and Georgetown Billabong in Australia; we then describe WSIBal, a simulation model based on the same balance equations and describe its application to Lake Jasper over a 19-year period; finally, we describe R-WSIBal, which evolved from WSIBal, and explain its application to 2500 km of the Murray River in Australia.

Impact of Navier’s Slip and MHD on a Hybrid Nanofluid Flow over a Porous Stretching/Shrinking Sheet with Heat Transfer

The main objective of this study is to explore the inventive conception of the magnetohydrodynamic flow of a hybrid nanofluid over-porous stretching/shrinking sheet with the effect of radiation and mass suction/injection. The hybrid nanofluid advances both the manufactured nanofluid of the current region and the base fluid. For the current investigation, hybrid nanofluids comprising two different kinds of nanoparticles, aluminium oxide and ferrofluid, contained in water as a base fluid, are considered. A collection of highly nonlinear partial differential equations is used to model the whole physical problem. These equations are then transformed into highly nonlinear ordinary differential equations using an appropriate similarity technique. The transformed differential equations are nonlinear, and thus it is difficult to analytically solve considering temperature increases. Then, the outcome is described in incomplete gamma function form. The considered physical parameters namely, magnetic field, Inverse Darcy number, velocity slip, suction/injection, temperature jump effects on velocity, temperature, skin friction and Nusselt number profiles are reviewed using plots. The results reveal that magnetic field, and Inverse Darcy number values increase as the momentum boundary layer decreases. Moreover, higher values of heat sources and thermal radiation enhance the thermal boundary layer. The present problem has various applications in manufacturing and technological devices such as cooling systems, condensers, microelectronics, digital cooling, car radiators, nuclear power stations, nano-drag shipments, automobile production, and tumour treatments.

The Exact Density of the Eigenvalues of the Wishart and Matrix-Variate Gamma and Beta Random Variables

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.

A METHOD FOR SYNTHESIZING TOPOS OF HASHES IN THE CATEGORICAL cN-SCHEME OF THE MARKOV ALGORITHM. Part 1

In organizational systems, economic interactions are studied. The purpose of this study is to synthesize a problem-oriented management subsystem for combating money laundering and terrorist financing. To achieve this goal, the problem of synthesizing a constructive way of processing such interactions is solved. The method is based on the development of the N-scheme of the Markov algorithm with an output that cannot be considered normal according to Markov. This algorithm scheme allows us to consider a quadruple consisting of words in the ordered and extended Markov alphabet A2, morphism numbers N, source numbers i, and the alphabet M. From the N-scheme of the Markov algorithm, its categorical version is synthesized. The need for this scheme of the Markov algorithm is due to the solution of the problem of folding Markov occurrences of words into each other of large length for their more compact representation. The idea of organizing a convolution for the entire topos is tested separately. The term “topos of hashes” is introduced into scientific circulation. This topos is considered as an alternative way to represent blocks of chains of Markov occurrences. The hash topos is formed from hash trees known in blockchain technology. This idea follows from previous studies of classifiers in topos, as well as the possibility of splitting chains of Markov occurrences into blocks. The convolution is considered using gamma and hash functions. In the course of the study, it was possible to theoretically show the connection between Markov algorithms and blockchain technology, taking into account the fact that each chain of occurrences of words in each other is considered as a category, and these chains are formed in different sources, otherwise worlds, and the way they are synthesized may have signs of a creative process, then a connection was found between the theories of categories, algorithms, blockchain technology and intuitionistic models by S. Kripke and L. E. Ya. Brouwer.