COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITELY MANY GAMMA FUNCTIONS

In this paper,by deriving an inequality involving the generating function of the Bernoulli numbers,the author introduces a new ratio of finitely many gamma functions,finds complete monotonicity of the second logarithmic derivative of the ratio,and simply reviews the complete monotonicity of several linear combinations of finitely many digamma or trigamma functions.

Gamma Functions and Siegel Integrals on Nonselfdual Cones

In this paper,we will define Gamma functions and Siegel integrals on nonselfdual cones and give some applications.

The Infinite Polynomial Products of the Gamma and Zeta Functions

Starting with the binomial coefficient and using its infinite product representation, the infinite product representation of the gamma function and of the zeta function are composed of an exponential and of a trigonometric component and proved. It is proved, that all these components define imaginary roots on the critical line, if written in the form as they are in the functional equation of the zeta function.

Probability Laws Derived from the Gamma Function

Several densities or probability laws of continuous random variables derive from the Euler Gamma function. These laws form the basis of sampling theory, namely hypothesis testing and estimation. Namely the gamma, beta, and Student law, through the chi-square law and the normal law are all distributions resulting from applications of Euleur functions.

NONEXISTENCE IN POWER AND GAMMA DENSITY REGRESSION, SUM OF NONNEGATIVE TERMS

When we use the power function a(c + x)b and gamma density axbe-cx to fit the data by the least squares method, we have to address the question of existence. The closure of the set of each type of these functions defined on a finite domain is determined. We derive a way to determine the closure of a sum of nonnegative functions if the closures of the summands are available.

Evaluation of Generalized Error Function via Fast-Converging Power Series

A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.

Consequences of Invariant Functions for the Riemann Hypothesis

This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specifically the Robin inequality and the Riemann hypothesis. The exploration of using invariant properties of these functions to derive insights into twin primes and sequential primes is a potentially innovative concept that deserves careful consideration by the mathematical community.

Volume of unit ball in an n-dimensional normed space and its asymptotic properties

Let Bp^n={x∈R^b｜‖x‖p≤1} be the unit ball of p norm in the n-dimensional normed space εp＆n.The formula for the volume of Bp^n was obtained and its asymptotic properties were found out as n→∞and p→∞.

Some properties of the psi function and evaluations of γ

In this paper, the authors show some monotonicity and concavity of the classical psi function, by which several known results are improved and some new asymptotically sharp estimates are obtained for this function. In addition, applying the new results to the psi function, the authors improve the well-known lower and upper bounds for the approximate evaluation of Euler＇s constant γ.

APPLICATIONS OF HYPERGEOMETRIC SUMMATION THEOREMS OF KUMMER AND DIXON INVOLVING DOUBLE SERIES

Using series iteration techniques identities and apply each of these identities in we derive a number of general double series order to deduce several hypergeometric reduction formulas involving the Srivastava-Daoust double hypergeometric function. The results presented in this article are based essentially upon the hypergeometric summation theorems of Kummer and Dixon.

Determinantal Expressions and Recursive Relations for the Bessel Zeta Function and for a Sequence Originating from a Series Expansion of the Power of Modified Bessel Function of the First Kind

In the paper,by virtue of a general formula for any derivative of the ratio of two differentiable functions,with the aid of a recursive property of the Hessenberg determinants,the authors establish determinantal expressions and recursive relations for the Bessel zeta function and for a sequence originating from a series expansion of the power of modified Bessel function of the first kind.

ALGEBRAIC DIFFERENTIAL INDEPENDENCE CONCERNING THE EULER Γ-FUNCTION AND DIRICHLET SERIES

This article investigates the algebraic differential independence concerning the Euler Γ-function and the function F in a certain class F which contains Dirichlet L-functions,L-functions in the extended Selberg class, or some periodic functions. We prove that the EulerΓ-function and the function F cannot satisfy any nontrivial algebraic differential equations whose coefficients are meromorphic functions Ø with ρ(Ø) < 1.

ON THE DIFFERENTIAL AND DIFFERENCE INDEPENDENCE OFΓANDζ

In this paper,we study the algebraic differential and the difference independence between the Riemann zeta function and the Euler gamma function.It is proved that the Riemann zeta function and the Euler gamma function cannot satisfy a class of nontrivial algebraic differential equations and algebraic difference equations.

On Computation Formula for a Series

It is one of the most interesting problems in number theory to compute someespecial series by using Zeta and Gamma functions, and results have been obtained for someespecial series. In this paper, we give an important formula which is proved also by usingZeta and Gamma functions.

Evaluation of Exponential Integral by Means of Fast-Converging Power Series

Exponential integral for real arguments is evaluated by employing a fast-converging power series originally developed for the resolution of Grandi’s paradox. Laguerre’s historic solution is first recapitulated and then the new solution method is described in detail. Numerical results obtained from the present series solution are compared with the tabulated values correct to nine decimal places. Finally, comments are made for the further use of the present approach for integrals involving definite functions in denominator.

Study on gamma response function of EJ301 organic liquid scintillator with GEANT4 and FLUKA

The gamma response function is required for energy calibration of EJ301 （5 cm in diameter and 20 cm in height） organic liquid scintillator detector by means of gamma sources. The GEANT4 and FLUKA Monte Carlo simulation packages were used to simulate the response function of the detector for standard 22Na, 60Co, 137Cs gamma sources. The simulated results showed a good agreement with experimental data by incorporating the energy resolution function to simulation codes. The energy resolution and the position of the maximum Compton electron energy were obtained by comparing measured light output distribution with simulated one. The energy resolution of the detector varied from 21.2% to 12.4% for electrons in the energy region from 0.341 MeV to 1.12 MeV. The accurate position of the maximum Compton electron energy was determined at the position 81% of maximum height of Compton edges distribution. In addition, the relation of the electron energy calibration and the effective neutron detection thresholds were described in detail. The present results indicated that both packages were suited for studying the gamma response function of EJ301 detector.

A Two-Sided Inequality of Gamma Function

This note shows that the inequaiity （x/e）^x√2πx（1＋1/12x）〈Г（x＋1）〈（x/e）^x√2πx（1＋1/12x-0.5） holds for all x≥1.

Difference Independence of the Euler Gamma Function

In this paper,the authors established a sharp version of the difference analogue of the celebrated Holder’s theorem concerning the differential independence of the Euler gamma functionГ.More precisely,if P is a polynomial of n+1 variables in C[X,Y0,…,Yn-1]such that P(s,Г(s+a0),…,Г(s+an-1))≡0 for some(a0,…,an-1)∈Cn and ai-aj■Z for any 0≤i<j≤n-1,then they have P≡0.Their result complements a classical result of algebraic differential independence of the Euler gamma function proved by H?lder in 1886,and also a result of algebraic difference independence of the Riemann zeta function proved by Chiang and Feng in 2006.

On Algebraic Methods for the Riemann Hypothesis

Properties of the gamma function are examined with implications for the Riemann hypothesis. Some new relations are obtained for the roots of the Zeta function using the properties of the Gamma function, the Bernoulli function, and limitations imposed by the rational relations of the complex roots of the Riemann zeta function, and the rational relations of the complex Gamma functions.

Analysis of the Microphysical Properties of a Stratiform Rain Event Using an L-Band Profiler Radar

This paper investigates spatial and temporal distributions of the microphysical properties of precipitating stratiform clouds based on Doppler spectra of rain particles observed by an L-band profiler radar.The retrieval of raindrop size distributions（RSDs） is accomplished through eliminating vertical air motion and isolating the terminal fall velocity of raindrops in the observed Doppler velocity spectrum.The microphysical properties of raindrops in a broad stratiform region with weak convective cells are studied using data collected from a 1320-MHz wind profiler radar in Huayin,Shaanxi Province on 14 May 2009.RSDs and gamma function parameters are retrieved at altitudes between 700 and 3000 m above the surface,below a melting layer.It is found that the altitude of the maximum number of raindrops was closely related to the surface rain rate.The maximum number of large drops was observed at lower altitudes earlier in the precipitation event but at higher altitudes in later periods,suggesting decreases in the numbers of large and medium size raindrops.These decreases may have been caused by the breakup of larger drops and evaporation of smaller drops as they fell.The number of medium size drops decreased with increasing altitude.The relationship between reflectivity and liquid water content during this precipitation event was Z = 1.69×10~4M^（1.5）,and the relationship between reflectivity and rain intensity was Z = 256I^（1.4）.