Polygamma Research Articles

Asymptotic expansions of the logarithm of the gamma function in the terms of the polygamma functions

We present new asymptotic expansions of the logarithm of the gamma function in terms of the polygamma functions. Based on these expansions, we prove new complete mono- tonicity properties of some functions involving the gamma and polygamma functions. As con- sequences of them we establish new upper and lower bounds for the gamma function in terms of the polygamma functions.

A DEFINITE INTEGRAL FORMULA

A remarkably large number of integral formulas have been investigated and developed. Certain large number of integral formulas are expressed as derivatives of some known functions. Here we choose to recall such a formula to present explicit expressions in terms of Gamma function, Psi function and Polygamma functions. Some simple interesting special cases of our main formulas are also considered. It is also pointed out that the same argument can establish explicit integral formulas for other those expressed in terms of derivatives of some known functions.

Asymptotic expansions of the gamma function and Wallis function through polygamma functions

A new view on classical asymptotic expansions of the logarithm of gamma function is given. Then general formulae for the asymptotic expansions of the logarithm of gamma function and the Wallis power function through polygamma functions are derived and analysed.

A completely monotonic function involving the tri- and tetra-gamma functions

Abstract The di-gamma function ψ(x) is defined on (0,∞) by $\psi (x) = \frac{{\Gamma '(x)}} {{\Gamma (x)}} $ and ψ (i)(x) for i ∈ ℕ denote the polygamma functions, where Γ(x) is the classical Euler’s gamma function. In this paper we prove that a function involving the difference between [ψ′(x)]2 + ψ″(x) and a proper fraction of x is completely monotonic on (0,∞).

Exponential, gamma and polygamma functions: Simple proofs of classical and new inequalities

Some inequalities for the exponential, gamma and polygamma functions are established by means of the mean value theorem of differential calculus. In particular, the cases of the ratio of gamma and polygamma functions are considered. In some cases the results are new, in some others, classical results are recovered or extended.

Certain relationships among polygamma functions, Riemann zeta function and generalized zeta function

Many useful and interesting properties, identities, and relations for the Riemann zeta function and the Hurwitz zeta function have been developed. Here, we aim at giving certain (presumably) new and (potentially) useful relationships among polygamma functions, Riemann zeta function, and generalized zeta function by modifying Chen’s method. We also present a double inequality approximating by a more rapidly convergent series. MSC:11M06, 33B15, 40A05, 26D07.

COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE DIVIDED DIFFERENCE OF PSI FUNCTIONS

AbstractNecessary and sufficient conditions are presented for a function involving the divided difference of the psi function to be completely monotonic and for a function involving the ratio of two gamma functions to be logarithmically completely monotonic. From these, some double inequalities are derived for bounding polygamma functions, divided differences of polygamma functions, and the ratio of two gamma functions.

Integral and series representations of the digamma and polygamma functions

We obtain a variety of series and integral representations of the digamma function $\psi(a)$. These in turn provide representations of the evaluations $\psi(p/q)$ at rational argument and for the polygamma function $\psi^{(j)}$. The approach is through a limit definition of the zeroth Stieltjes constant $\gamma_0(a)=-\psi(a)$. Several other results are obtained, including product representations for $\exp[\gamma_0(a)]$ and for the Gamma function $\Gamma(a)$. In addition, we present series representations in terms of trigonometric integrals Ci and Si for $\psi(a)$ and the Euler constant $\gamma=-\psi(1)$.

Single-valued harmonic polylogarithms and the multi-Regge limit

Abstract We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar $ \mathcal{N}=4 $ super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, (w, w ∗). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables (ν, n) which are related to (w, w ∗) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν, n) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to N3LLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.

Complete monotonicity of two functions involving the tri-and tetra-gamma functions

The psi function ψ(x) is defined by ψ(x) = Γ′(x)/Γ(x) and ψ (i) (x), for i ∈ ℕ, denote the polygamma functions, where Γ(x) is the gamma function. In this paper, we prove that the functions \( [\psi '(x)]^2 + \psi ''(x) - \frac{{x^2 + 12}} {{12x^4 (x + 1)^2 }} \)and \( \frac{{x + 12}} {{12x^4 (x + 1)}} - \{ [\psi '(x)]^2 + \psi ''(x)\} \) are completely monotonic on (0,∞).

Refinements of lower bounds for polygamma functions

In the paper, some lower bounds for polygamma functions are refined.

A class of logarithmically completely monotonic functions related to the gamma function with applications

In this article, a necessary and sufficient condition and a necessary condition are established for a class of functions involving the gamma function to be logarithmically completely monotonic on . As applications of the necessary and sufficient condition, several two-sided bounding inequalities for the psi and polygamma functions and the ratio of two gamma functions are derived.

ANALOGUES OF A TRANSFORMATION FORMULA OF RAMANUJAN

We derive two new analogues of a transformation formula of Ramanujan involving the Gamma function and Riemann zeta function present in his Lost Notebook. Both involve infinite series consisting of Hurwitz zeta functions and yield modular-type relations. As a special case of the first formula, we obtain an identity involving polygamma functions given by A. P. Guinand and as a limiting case of the second formula, we derive the transformation formula of Ramanujan.

On the Empirical-Statistical Modeling of SAR Images With Generalized Gamma Distribution

In this paper, an efficient statistical model, called generalized Gamma distribution (GΓD), for the empirical modeling of synthetic aperture radar (SAR) images is proposed. The GΓD forms a large variety of alternative distributions (especially including Rayleigh, exponential, Nakagami, Gamma, Weibull, and log-normal distributions commonly used for the probability density function (pdf) of SAR images as special cases), and is flexible to model the SAR images with different land-cover typologies. Moreover, based on second-kind cumulants, a closed-form estimator for GΓD parameters is derived by exploiting the second-order approximation for Polygamma function. Without involving the numerical iterative process for solutions, this estimator is computationally efficient and, hence, can make the GΓD convenient for applications in the online SAR image processing. Finally, experimental results from tests carried out with actual SAR images demonstrate that the GΓD can achieve better goodness of fit than the state-of-the-art pdfs.

Krull’s theory for the double gamma function

The Barnes’ G -function G ( x ) = 1/ Γ 2 , satisfies the functional equation log G ( x + 1) − log G ( x ) = log Γ ( x ). We complement W. Krull’s work in Bemerkungen zur Differenzengleichung g ( x + 1) − g ( x ) = φ ( x ), Math. Nachrichten 1 (1948), 365–376 with additional results that yield a different characterization of the function G , new expansions and sharp bounds for G on x > 0 in terms of Gamma and Digamma functions, a new expansion for the Gamma function and summation formulae with Polygamma functions.

Transcendence of multi-indexed infinite series

We consider the transcendence of the multi-indexed series ∑ n 1 , … , n k = 1 ∞ f ( n 1 , … , n k ) n 1 ⋯ n k and then extend our results to series of the form ∑ n 1 , … , n k = 0 ∞ f ( n 1 , … , n k ) A 1 ( n 1 ) ⋯ A k ( n k ) B 1 ( n 1 ) ⋯ B k ( n k ) where f is a k-periodic function and each A i ( x ) , B i ( x ) is a polynomial with algebraic coefficients. We relate these series to special values of the digamma function as well as polygamma functions. We also show some conditional implications on the transcendence of the series via Schanuel's Conjecture.

Complete monotonicity results for some functions involving the gamma and polygamma functions

In this paper we prove new complete monotonicity properties of some functions involving the gamma function. As consequences of them we establish various new upper and lower bounds for the gamma function and the harmonic numbers.

A class of two-sided inequalities involving the psi and polygamma functions

A class of two-sided inequalities involving the psi and polygamma functions is presented. These inequalities provide new bounds for the psi function in terms of the polygamma functions.

A main inequality for several special functions

By using a recent generalization of the Cauchy–Schwarz inequality we obtain several new inequalities for some basic special functions such as beta and incomplete beta functions, gamma, polygamma and incomplete gamma functions, error functions, various kinds of Bessel functions, exponential integral functions, Gauss hypergeometric and confluent hypergeometric functions, elliptic integrals, moment generating functions and the Riemann zeta function. We also apply the generalized Cauchy–Schwarz inequality to some classical integral transforms. All these new inequalities obey the general form f 2 ( x ) ≤ k ( x ) f ( p x + q ) f ( ( 2 − p ) x − q ) , in which p , q are real parameters and k ( x ) is a specific positive function.

An exact fluctuating 1/2-BPS configuration

This work explores the role of thermodynamic fluctuations in the two parameter giant and superstar configurations characterized by an ensemble of arbitrary liquid droplets or irregular shaped fuzzballs. Our analysis illustrates that the chemical and state-space geometric descriptions exhibit an intriguing set of exact pair correction functions and the global correlation lengths. The first principle of statistical mechanics shows that the possible canonical fluctuations may precisely be ascertained without any approximation. Interestingly, our intrinsic geometric study exemplifies that there exist exact fluctuating 1/2-BPS statistical configurations which involve an ensemble of microstates describing the liquid droplets or fuzzballs. The Gaussian fluctuations over an equilibrium chemical and state-space configurations accomplish a well-defined, non-degenerate, curved and regular intrinsic Riemannian manifolds for all physically admissible domains of black hole parameters. An explicit computation demonstrates that the underlying chemical correlations involve ordinary summations, whilst the state-space correlations may simply be depicted by standard polygamma functions. Our construction ascribes definite stability character to the canonical energy fluctuations and to the counting entropy associated with an arbitrary choice of excited boxes from an ensemble of ample boxes constituting a variety of Young tableaux.