Polygamma Research Articles

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.

On the Sums over Inverse Powers of Zeros of the Hurwitz Zeta Function and Some Related Properties of These Zeros

Recently, we have applied the generalized Littlewood theorem concerning contour integrals of the logarithm of the analytical function to find the sums over inverse powers of zeros for the incomplete gamma and Riemann zeta functions, polygamma functions, and elliptical functions. Here, the same theorem is applied to study such sums for the zeros of the Hurwitz zeta function ζ(s,z), including the sum over the inverse first power of its appropriately defined non-trivial zeros. We also study some related properties of the Hurwitz zeta function zeros. In particular, we show that, for any natural N and small real ε, when z tends to n = 0, −1, −2… we can find at least N zeros of ζ(s,z) in the ε neighborhood of 0 for sufficiently small |z+n|, as well as one simple zero tending to 1, etc.

Series involving polygamma functions and certain variant Euler harmonic sums

Series involving polygamma functions and certain variant Euler harmonic sums

Corrigendum to “A harmonic mean inequality for the polygamma function"

Corrigendum to “A harmonic mean inequality for the polygamma function"

Several Functions Originating from Fisher–Rao Geometry of Dirichlet Distributions and Involving Polygamma Functions

In this paper, the authors review and survey some results published since 2020 about (complete) monotonicity, inequalities, and their necessary and sufficient conditions for several newly introduced functions involving polygamma functions and originating from the estimation of the sectional curvature of the Fisher–Rao geometry of the Dirichlet distributions in the two-dimensional case.

Pole decomposition of BFKL eigenvalue at zero conformal spin and the real part of digamma function

We consider the powers of leading order eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation at zero conformal spin. Using reflection identities of harmonic sums we demonstrate how involved generalized polygamma functions are introduced by pole separation of a rather simple digamma function. This generates higher weight generalized polygamma functions at any given order of perturbative expansion. As a byproduct of our analysis we develop a general technique for calculating powers of the real part of digamma function in a pole separated form.

Statistics of electron-multiplying charge-coupled devices

Electron-multiplying charge-coupled devices are efficient imaging devices for low-surface-brightness ultraviolet astronomy from space. The large amplification allows photon counting (PC), the detection of events versus nonevents. This paper provides the statistics of the observation process, the photon-counting process, the amplification process, and the compression. The expression for the signal-to-noise of PC is written in terms of the polygamma function. The optimal exposure time is a function of the clock-induced charge. The exact distribution of amplification process is a simple-to-compute powered matrix. The optimal cutoff for comparing to the read noise is close to a strong function of the read noise and a weak function of the electron-multiplying gain and photon rate. A formula gives the expected compression rate.

Complete monotonicity involving the divided difference of polygamma functions

For r, s ? R and ? = min {r, s}, let D[x + r, x + s; ?n?1] ? ??n (x) be the divided difference of the functions ?n?1 = (?1)n ?(n?1) (n ? N) on (??,?), where ?(n) stands for the polygamma functions. In this paper, we present the necessary and sufficient conditions for the functions x ? ?k i=1 ?mi (x) ? ?k ?k i=1 ?ni (x) , x ? ?k i=1 ?ni (x) ? ?k?snk (x) to be completely monotonic on (??,?), where mi, ni ? N for i = 1,..., k with k ? 2 and snk = ?k i=1 ni. These generalize known results and gives an answer to a problem.

A generic inequality for basic special functions

We introduce a generic inequality of the form and show that many special functions such as gamma and polygamma functions, Riemann zeta function, beta and incomplete beta functions, Gauss and confluent hypergeometric functions, elliptic integrals and error function satisfy such type of the inequality. We also obtain new inequalities of this type for inverse trigonometric functions and .

Exact ground state energy of a system with an arbitrary number of dipoles at the sites of a regular one-dimensional crystal lattice

We consider a finite classical system of electric dipoles localized at the sites of a regular one-dimensional crystal lattice. It is shown that the ground state energy of such a system can be calculated exactly for an arbitrary number of dipoles. The expression for the ground state energy can be given in terms of special functions such as Riemann zeta functions and polygamma functions for any arbitrary number of dipoles, and reduces to the correct result in the thermodynamic limit as the number of dipoles tends to infinity. Simpler, but very accurate, approximate expressions for the ground state energy are also introduced.

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.

Existence and stability of symmetric and asymmetric patterns for the half-Laplacian Gierer–Meinhardt system in one-dimensional domain

In this paper, we study the existence and stability of multiple spikes pattern to the fractional Gierer–Meinhardt model with periodic boundary conditions and the fractional power [Formula: see text]. Specifically, we rigorously establish the existence of symmetric multiple spikes and asymmetric two-spikes solutions by the classical Lyapunov–Schmidt reduction method. We also investigate the stability of the constructed solution by studying its associated large and small eigenvalue problems, where we need to consider two nonlocal eigenvalue problems in their fractional versions. In the study of the large eigenvalue problem, the quantity [Formula: see text] is the critical threshold which determines the stability of [Formula: see text]-peaked solutions. For the symmetric two-spikes pattern we obtain the asymptotic expansion for the critical threshold [Formula: see text] up to the second order. Moreover, we provide some elementary properties of the Green’s function, including the first and second derivatives, they are linked to the location of the spikes and the stability. Among these properties on the Green’s function, we find out that the polygamma function [Formula: see text] plays a crucial role.

The Estimators of the Bent, Shape and Scale Parameters of the Gamma-Exponential Distribution and Their Asymptotic Normality

When modeling real phenomena, special cases of the generalized gamma distribution and the generalized beta distribution of the second kind play an important role. The paper discusses the gamma-exponential distribution, which is closely related to the listed ones. The asymptotic normality of the previously obtained strongly consistent estimators for the bent, shape, and scale parameters of the gamma-exponential distribution at fixed concentration parameters is proved. Based on these results, asymptotic confidence intervals for the estimated parameters are constructed. The statements are based on the method of logarithmic cumulants obtained using the Mellin transform of the considered distribution. An algorithm for filtering out unnecessary solutions of the system of equations for logarithmic cumulants and a number of examples illustrating the results obtained using simulated samples are presented. The difficulties arising from the theoretical study of the estimates of concentration parameters associated with the inversion of polygamma functions are also discussed. The results of the paper can be used in the study of probabilistic models based on continuous distributions with unbounded non-negative support.

One-loop gluon amplitudes in AdS

We initiate the study of one-loop gluon amplitudes in AdS space. These amplitudes were recently computed at tree level for a variety of backgrounds of the form AdSd+1× S3. For concreteness, we compute the one-loop correction to the massless gluon amplitude on AdS5×S3, which corresponds to the four-point correlator of the flavor current multiplet in the dual 4d mathcal{N} = 2 SCFT. This requires solving a mixing problem that involves tree-level amplitudes of arbitrarily massive Kaluza-Klein modes. The final answer has the same color structure as in flat space but the dependence on Mandelstam variables is more complicated, with logarithms replaced by polygamma functions.

Integral representations, monotonic properties, and inequalities on extended central binomial coefficient

In the paper, with the help of Kazarinoff's integral representation for the ratio of two gamma functions, with the aid the duplication formula of the digamma function, by virtue of integral representations of polygamma functions, and in the light of the L'Hôpital type monotonicity rule, the author describes extended binomial coefficients in terms of the gamma functions and the falling factorials, collects three integral representations of central binomial coefficients, establishes three integral representations of extended central binomial coefficients, proves decreasing and increasing properties of two power‐exponential functions involving extended (central) binomial coefficients, and derives several double and sharp inequalities for bounding extended (central) binomial coefficients, and compares these inequalities with known ones.

Decreasing properties of two ratios defined by three and four polygamma functions

In the paper, by virtue of the convolution theorem for the Laplace transforms, with the aid of three monotonicity rules for the ratios of two functions, of two definite integrals, and of two Laplace transforms, in terms of the majorization, and in the light of other analytic techniques, the author presents decreasing properties of two ratios defined by three and four polygamma 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.

Monotonic properties for ratio of the generalized (p,k)-polygamma functions

Monotonic properties for ratio of the generalized (p,k)-polygamma functions

Fibonacci-Zeta infinite series associated with the polygamma functions

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.

New summation formulas of Fox-Wright-type series containing the polygamma functions

The main focus of the present paper is to establish new summations formulas of Fox-Wright-type series containing the polygamma functions in terms of some special functions (such as the G-Meijer functions). A number of interesting new special cases and consequences of the main results are also considered. Moreover, we present some numerical results that illustrate the results of this work.