Ratio Of Gamma Functions Research Articles

Logarithmically complete monotonicity of reciprocal ARCTAN function

We prove the conjecture stated in F. Qi and R. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequal. Appl. (2019), that the function 1∕ arctan is logarithmically completely monotonic on (0,∞), but not a Stieltjes transform.

Complete monotonicity of the remainder of the asymptotic series for the ratio of two gamma functions

For u>0 with u≠1,2 and m∈N and letfm(x;u)=1u−1ln⁡Γ(x+u)Γ(x+1)−ln⁡(x+u2)−∑k=2mck(u)xk. We prove the complete monotonicity of fm(x;u) for m=2,3 and f4n−1(x;1/2), −f4n−2(x;1/2) for n∈N in x on (0,∞). From these several new sharp bounds for the ratio of gamma functions and divided difference of polygamma functions are established. Lastly, a conjecture is posed.

A combinatorial proof of the Gaussian product inequality beyond the MTP2 case

Abstract A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector X = ( X 1 , … , X d ) {\boldsymbol{X}}=\left({X}_{1},\ldots ,{X}_{d}) of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on X {\boldsymbol{X}} is shown to be strictly weaker than the assumption that the density of the random vector ( ∣ X 1 ∣ , … , ∣ X d ∣ ) \left(| {X}_{1}| ,\ldots ,| {X}_{d}| ) is multivariate totally positive of order 2, abbreviated MTP 2 {\text{MTP}}_{2} , for which the GPI is already known to hold. Under this condition, the paper highlights a new link between the GPI and the monotonicity of a certain ratio of gamma functions.

A class of completely monotonic functions involving the polygamma functions

Let Gamma (x) denote the classical Euler gamma function. We set psi _{n}(x)=(-1)^{n-1}psi ^{(n)}(x) (nin mathbb{N}), where psi ^{(n)}(x) denotes the nth derivative of the psi function psi (x)=Gamma '(x)/Gamma (x). For λ, α, beta in mathbb{R} and m,nin mathbb{N}, we establish necessary and sufficient conditions for the functions L(x;λ,α,β)=ψm+n(x)−λψm(x+α)ψn(x+β)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ L(x;\\lambda ,\\alpha ,\\beta )=\\psi _{m+n}(x)-\\lambda \\psi _{m}(x+ \\alpha ) \\psi _{n}(x+\\beta ) $$\\end{document} and -L(x;lambda ,alpha ,beta ) to be completely monotonic on (-min (alpha ,beta ,0),infty ).As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions.

An Integral Involving a Generalized Hypergeometric Function

In 1961, MacRobert in his very popular, useful and interesting research paper obtained a new type of finite integrals and used the integrals to evaluate integral involving E-functions which he had developed and is a generalization of hypergeometric and generalized hypergeometric functions. The main objective of this short research paper is to find an exciting integral associated with a generalized hypergeometric function by using the integrals obtained by MacRobert. The beauty of our results is that they appear on the product of the ratios of gamma functions. It is clear that the integral associated with gamma functions, the results are very useful from the perspective of the point of view of applications. In terms of parameters, one can easily derive the known integrals due to Rathie and the integral given in Mathai and Saxena's book. It is no exaggeration to mention here that, for other integrals, the transformation and summation formulas involve generalized hypergeometric function.

New Improved Inequalities Associated with Ratios of Gamma Functions

In this paper, we improve the estimate for the products ∏ k = 1 n 4 k − 3 / 4 k and ∏ k = 1 n 4 k − 1 / 4 k , obtained by Cringanu “inequalities associated with ratios of gamma functions.”

Monotonicity Properties Related to the Ratio of Two Gamma Functions

In this paper we investigate the monotonicity properties related to the ratio of gamma functions, from which some related asymptotics and inequalities are established. Some special cases also confirm the conjectures of C.-P. Chen (Appl Math Comput 283:385–396, 2016).

Nielsen's beta function and some infinitely divisible distributions

AbstractWe show that a large collection of special functions, in particular Nielsen's beta function, are generalized Stieltjes functions of order 2, and therefore logarithmically completely monotonic. This includes the Laplace transform of functions of the form , where f is itself the Laplace transform of a sum of dilations and translations of periodic functions. Our methods are also applied to ratios of Gamma functions, and to the remainders in asymptotic expansions of the double Gamma function of Barnes.

From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions

In this paper, the authors review origins, motivations, and generalizations of a series of inequalities involving finitely many exponential functions and sums. They establish three new inequalities involving finitely many exponential functions and sums by finding convexity of a function related to the generating function of the Bernoulli numbers. They also survey the history, backgrounds, generalizations, logarithmically complete monotonicity, and applications of a series of ratios of finitely many gamma functions, present complete monotonicity of a linear combination of finitely many trigamma functions, construct a new ratio of finitely many gamma functions, derive monotonicity, logarithmic convexity, concavity, complete monotonicity, and the Bernstein function property of the newly constructed ratio of finitely many gamma functions. Finally, they suggest two linear combinations of finitely many trigamma functions and two ratios of finitely many gamma functions to be investigated.

Completely monotonicity of class functions involving the polygamma and related functions

In this paper, the authors prove some inequalities and completely monotonic properties of polygamma functions. As an application, we give lower bound for the zeta function on natural numbers. Partially, we answer the fifth and sixth open problems listed in [F. Qi and R. P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequalities Appl. 2019(36) (2019) 42]. We propose two open problems on completely monotonic functions related to polygamma functions.

Three-point functions at strong coupling in the BMN limit

We consider structure constants of single-trace operators at strong coupling in planar mathcal{N} = 4 SYM theory using the hexagon formalism. We concentrate on heavy-heavy- light correlators where the heavy operators are BMN operators, with large R-charges and finite anomalous dimensions, and the light one is a finite-charge chiral primary operator. They describe the couplings between two highly boosted strings and a supergravity mode in the bulk dual. In the hexagon framework, two sums over virtual magnons are needed to bind the hexagons together around the light operator. We evaluate these sums explicitly at strong coupling, for a certain choice of BMN operators, and show that they factorise into a ratio of Gamma functions and a simple stringy prefactor. The former originates from giant mirror magnons scanning the AdS geometry while the latter stems from small fluctuations around the BMN vacuum. The resulting structure constants have poles at positions where an enhanced mixing with double-trace operators is expected and zeros whenever the process is forbidden by supersymmetry. We also discuss the transition to the classical regime, when the length of the light operator scales like the string tension, where we observe similitudes with the Neumann coefficients of the pp-wave String Field Theory vertex.

A Solution to Qi’s Conjecture on a Double Inequality for a Function Involving the Tri- and Tetra-Gamma Functions

In the paper, the author gives a solution to a conjecture on a double inequality for a function involving the tri- and tetra-gamma functions, which was first posed in Remark 6 of the paper “Complete monotonicity of a function involving the tri- and tetragamma functions” (2015) and repeated in the seventh open problem of the paper “On complete monotonicity for several classes of functions related to ratios of gamma functions” (2019).

Complete monotonicity and inequalites involving Gurland's ratios of gamma functions

Complete monotonicity and inequalites involving Gurland's ratios of gamma functions

INEQUALITIES ASSOCIATED WITH RATIOS OF GAMMA FUNCTIONS

We use properties of the gamma function to estimate the products$\prod _{k=1}^{n}(4k-3)/4k$and$\prod _{k=1}^{n}(4k-1)/4k$, motivated by the work of Chen and Qi [‘Completely monotonic function associated with the gamma function and proof of Wallis’ inequality’,Tamkang J. Math.36(4) (2005), 303–307] and Morticiet al.[‘Completely monotonic functions and inequalities associated to some ratio of gamma function’,Appl. Math. Comput.240(2014), 168–174].

Products of ratios of gamma functions — An application to the distribution of the test statistic for testing the equality of covariance matrices

An expansion of the ratio of two gamma functions is used to obtain an adequate representation for the product of ratios of gamma functions which enables the derivation of approximating expansions for the distribution of the test statistic used for testing the equality of several covariance matrices. These asymptotic approximations are based on mixtures of gamma distributions and are obtained using a new expression of the Fourier transform of the density function of the logarithm of the test statistic and also on a matching moments technique. Numerical studies show that these approximations are simple, precise and easy to implement. A computational module, developed for Mathematica software, is presented together with an example of application.

Complete monotonicity involving some ratios of gamma functions

In this paper, by using the properties of an auxiliary function, we mainly present the necessary and sufficient conditions for various ratios constructed by gamma functions to be respectively completely and logarithmically completely monotonic. As consequences, these not only unify and improve certain known results including Qi’s and Ismail’s conclusions, but also can generate some new inequalities.

Reciprocity theorems involving the q-gamma function

In one of his notebooks, Ramanujan stated but did not prove an identity for a ratio of gamma functions involving two separate variables. Such an identity was referred to as a reciprocity theorem due to the symmetry relations between the variables. The two variable result was later proved and generalized to several variables by Berndt and Koukoulopoulos (Proc Am Math Soc 601:2369–2373, 2007). In this paper, we state and prove similar reciprocity theorems for ratios of q-gamma functions involving two and more variables.

Testing the equality of several linear regression models

The linear regression models are widely used in different research fields, and often there is the need to analyze if there are similarities between two or more different linear models or to verify if a given relation between two variables remains the same in different intervals of time, in particular in cases where small differences might make a big difference. Motivated by these problems the authors consider a test of equality of k linear regression models which is a simultaneous test of equality of slopes, intercepts and variances. In order to overcome the extreme difficulties that exist in the use of the exact distribution of the likelihood ratio test (LRT) statistic and to make this test reliable and easy to use, we propose the use of near-exact distributions to approximate the distribution of the LRT statistic, under $$H_0$$ , in the balanced case, and of new asymptotic approximations for the unbalanced case. The near-exact approximations are built by approximating one factor of an adequate factorization of the characteristic function of the logarithm of the LRT statistic and may be easily implemented. The asymptotic approximations are developed using an expansion for the ratio of gamma functions. The quality of these approximations is analyzed and confirmed. Power studies are conducted in order to better assess the performance of the test. Finally to illustrate the applicability of the test we consider a real data set of gross domestic product at market prices and final consumption expenditure in European countries and one tests the existence of similarities between countries.

On (p, q) and (q, k)-extensions of a double-inequality bounding a ratio of Gamma functions

In this paper, the authors present the (p, q) and (q, k)-extensions of a double inequality involving a ratio of Gamma functions. The method is based on some monotonicity properties of certain functions associated with the (p, q) and (q, k)-extensions of the Gamma function.

Monotonicity of a mean related to polygamma functions with an application

Let \(\psi_{n}= ( -1 ) ^{n-1}\)\(\psi^{ ( n ) }\) (\(n=0,1,2,\ldots \)), where \(\psi^{ ( n ) }\) denotes the psi and polygamma functions. We prove that for \(n\geq0\) and two different real numbers a and b, the function $$ x\mapsto\psi_{n}^{-1} \biggl( \frac{\int_{a}^{b}\psi_{n}(x+t)\,dt}{b-a} \biggr) -x $$ is strictly increasing from \(( -\min ( a,b ) ,\infty ) \) onto \(( \min ( a,b ) , ( a+b ) /2 ) \), which generalizes a well-known result. As an application, the complete monotonicity for a ratio of gamma functions is improved.