Digamma Function Research Articles

Notes on three conjectures involving the digamma and generalized digamma functions

In the paper, we solve one conjecture on an inequality involving digamma function, an open problem, and a conjecture on monotonicity of functions involving generalized digamma function. We also prove a new inequality for digamma function.

Generalized Harmonic Numbers

This paper presents new formulae for the harmonic numbers of order $k$, $H_{k}(n)$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)$ and $S^m_{k}(n)$. I believe this new formula for $H_{k}(n)$ is an improvement over the digamma function, $\psi$, because it's simpler and it stems from Faulhaber's formula, which provides a closed-form for the sum of powers of the first $n$ positive integers. We demonstrate how to create an exact power series for the harmonic numbers, a new integral representation for $\zeta(2k+1)$ and a new generating function for $\zeta(2k+1)$, among many other original results. The approaches and formulae discussed here are entirely different from solutions available in the literature.

Generalized Harmonic Numbers

This paper presents new formulae for the harmonic numbers of order $k$, $H_{k}(n)$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)$ and $S^m_{k}(n)$. I believe this new formula for $H_{k}(n)$ is an improvement over the digamma function, $\psi$, because it's simpler and it stems from Faulhaber's formula, which provides a closed-form for the sum of powers of the first $n$ positive integers. We demonstrate how to create an exact power series for the harmonic numbers, a new integral representation for $\zeta(2k+1)$ and a new generating function for $\zeta(2k+1)$, among many other original results. The approaches and formulae discussed here are entirely different from solutions available in the literature.

Some monotonicity properties and inequalities for the generalized digamma and polygamma functions

Several monotonicity and concavity results related to the generalized digamma and polygamma functions are presented. This extends and generalizes the main results of Qi and Guo and others.

Monotonicity, concavity, and inequalities related to the generalized digamma function

In this paper, we establish a concave theorem and some inequalities for the generalized digamma function. Hence, we give complete monotonicity property of a determinant function involving all kinds of derivatives of the generalized digamma function.

Complete Monotonicity of a Function Involving the (P, K)-Digamma Function

Complete Monotonicity of a Function Involving the (P, K)-Digamma Function

A Class of Simple Rearrangements of the Alternating Harmonic Series

We present an easily defined countable family of permutations of the natural numbers for which explicit rearrangements (i.e., the sums induced by the permutations) can be computed. The digamma function proves to be the key tool for the computations found here for the alternating harmonic series. The permutations φ under consideration are simple in the sense that φ○φ is the identity function. We show that the countable set of rearrangements obtained from the permutations considered below is dense in the reals.

Inequalities for the inverses of the polygamma functions

We provide an elementary proof of the left-hand side of the following inequality and give a new upper bound for it. $$\begin{aligned} \bigg [\frac{n!}{x-(x^{-1/n}+\alpha )^{-n}}\bigg ]^{\frac{1}{n+1}}&<((-1)^{n-1}\psi ^{(n)})^{-1}(x) \\&<\bigg [\frac{n!}{x-(x^{-1/n}+\beta )^{-n}}\bigg ]^{\frac{1}{n+1}}, \end{aligned}$$ where $$\alpha =[(n-1)!]^{-1/n}$$ and $$\beta =[n!\zeta (n+1)]^{-1/n}$$ , which was proved in Batir (J Math Anal Appl 328:452–465, 2007), and we prove the following inequalities for the inverse of the digamma function $$\psi $$ . $$\begin{aligned} \frac{1}{\log (1+e^{-x})}<\psi ^{-1}(x)< e^{x}+\frac{1}{2}, \quad x\in \mathbb {R}. \end{aligned}$$ The proofs are based on nice applications of the mean value theorem for differentiation and elementary properties of the polygamma functions.

Variations on a hypergeometric theme

The question was asked: Is it possible to express the function \begin{equation} \tag{1.1} h(a)\equiv\,{_4F_3}(a,a,a,a;2a,a+1,a+1;1) \label{question} \end{equation} in closed form? After considerable analysis, the answer appears to be "no", but during the attempt to answer this question, a number of interesting (and unexpected) related results were obtained, either as specialized transformations, or as closed-form expressions for several related functions. The purpose of this paper is to record and review both the methods attempted and the related identities obtained (specifically new $_4F_3(1)$, $_5F_6(1)$ and (generalized Euler) sums containing digamma functions) - the former for their educational merit, since they appear to be not-very-well-known, the latter because they do not appear to exist in the literature.

Zeros of the digamma function and its Barnes G-function analogue

ABSTRACTThe zeros of the digamma function are known to be simple and real, but up to now few identities involving them have appeared in the literature. By establishing a Weierstrass infinite product for a particular regularization of the digamma function, we are able to find interesting formulas for the sums of the nth powers of the reciprocals of its zeros, for . We make a parallel study of the zeros of the logarithmic derivative of the Barnes G-function. We also compare asymptotic estimates of the zeros of the digamma function and those of its Barnes G-function analogue.

Spatial and Social Paradigms for Interference and Coverage Analysis in Underlay D2D Network

The homogeneous Poisson point process is widely used to model spatial distribution of base stations and mobile terminals. The same process can be used to model underlay device-to-device (D2D) network; however, neglecting homophilic relation for D2D pairing presents underestimated system insights. In this paper, we model both spatial and social distributions of interfering D2D nodes as proximity-based independently marked homogeneous Poisson point process. The proximity considers physical distance between D2D nodes whereas social relationship is modeled as Zipf-based marks. We apply these two paradigms to analyze the effect of interference on coverage probability of distance-proportional power-controlled cellular user. Effectively, we apply two types of functional mappings (physical distance, social marks) to Laplace functional of PPP. The resulting coverage probability has no closed-form expression; however, for a subset of social marks, the mark summation converges to digamma and polygamma functions. This subset constitutes the upper and lower bounds on coverage probability. We present numerical evaluation of these bounds on coverage probability by varying number of different parameters. The results show that by imparting simple power control on cellular user, ultradense underlay D2D network can be realized without compromising the coverage probability of cellular user.

A harmonic mean inequality for the digamma function and related results

We present some inequalities and a concavity property of the digamma function \psi=\Gamma'/\Gamma , where \Gamma denotes Euler's gamma function. In particular, we offer a new characterization of Euler's constant \gamma=0.57721\dots . We prove that -\gamma is the minimum of the harmonic mean of \psi(x) and \psi(1/x) for x&gt;0 .

Generalized harmonic number summation formulae via hypergeometric series and digamma functions

By computing the derivatives of five classical hypergeometric summation theorems, and applying the related properties of the digamma function, we derive a large number of closed summation formulae for generalized harmonic numbers.

Exact and Approximate Capacitance and Force Expressions for the Electrostatic Interaction Between Two Equal-Sized Charged Conducting Spheres

We analyze the electrostatic interaction between two equal-sized charged conducting spheres. We obtain exact closed-form expressions for the capacitance coefficients and the electrostatic force in terms of the special q -digamma function. Additionally, we provide simpler-to-use approximate expressions for the capacitance coefficients and the force between the spheres, and then compare the approximations with the exact results.

A fresh approach to classical Eisenstein series and the newer Hilbert–Eisenstein series

This paper is concerned with new results for the circular Eisenstein series [Formula: see text] as well as with a novel approach to Hilbert–Eisenstein series [Formula: see text], introduced by Michael Hauss in 1995. The latter turns out to be the product of the hyperbolic sinh function with an explicit closed form linear combination of digamma functions. The results, which include differentiability properties and integral representations, are established by independent and different argumentations. Highlights are new results on the Butzer–Flocke–Hauss Omega function, one basis for the study of Hilbert–Eisenstein series, which have been the subject of several recent papers.

Exact Frequency Equation of a Linear Structure Carrying Lumped Elements Using the Assumed Modes Method

Combined systems consisting of linear structures carrying lumped attachments have received considerable attention over the years. In this paper, the assumed modes method is first used to formulate the governing equations of the combined system, and the corresponding generalized eigenvalue problem is then manipulated into a frequency equation. As the number of modes used in the assumed modes method increases, the approximate eigenvalues converge to the exact solutions. Interestingly, under certain conditions, as the number of component modes goes to infinity, the infinite sum term in the frequency equation can be reduced to a finite sum using digamma function. The conditions that must be met in order to reduce an infinite sum to a finite sum are specified, and the closed-form expressions for the infinite sum are derived for certain linear structures. Knowing these expressions allows one to easily formulate the exact frequency equations of various combined systems, including a uniform fixed–fixed or fixed-free rod carrying lumped translational elements, a simply supported beam carrying any combination of lumped translational and torsional attachments, or a cantilever beam carrying lumped translational and/or torsional elements at the beam's tip. The scheme developed in this paper is easy to implement and simple to code. More importantly, numerical experiments show that the eigenvalues obtained using the proposed method match those found by solving a boundary value problem.

Some extensions of the Prabhu-Srivastava theorem involving the (p,q)-gamma function

In this paper, we obtain some limit formulas for derivatives of (p,q)-gamma function and (p,q)- digamma function at their poles. These limit formulas extend the Prabhu-Srivastava theorem involving gamma function and digamma function.

A certain class of approximations for the $q$-digamma function

In this paper, we derive a class of approximations of the $q$-digamma function $\psi _q(x)$. The infinite family \[ I_a(x;q)=\log [x+a]_q+\frac {q^x\log q}{1-q^x}-\bigg (\frac 12-a\bigg )H(q-1)\log q, \] $a\in [0,1]$; $q>0$, can be used as approximating functions for $\psi _q(x)$, where $[x]_q=(1-q^x)/(1-q)$ and $H(\cdot )$ is the Heaviside step function. We show that, for all $a\in [0,1]$, $I_a$ is asymptotically equivalent to $\psi _q(x)$ for $q>0$ and is a good pointwise approximation.

An inequality involving the gamma and digamma functions

Abstract In the paper, the authors establish an inequality involving the gamma and digamma functions and apply it to prove the negativity and monotonicity of a function involving the gamma and digamma functions.

Exact spatial density of ideal Bose atoms in a one-dimensional harmonic trap

We have proposed an exact analytical solution to the problem of Bose–Einstein condensation (BEC) of harmonically trapped, one-dimensional, and ideal atoms. It is found that the number of atoms in vapor is characterized by an analytical function, which involves a q -digamma function in mathematics. We employ the q -digamma function to calculate the spatial density n(z;T, N) of ideal Bose atoms in a one-dimensional harmonic trap. The first main finding in this paper is that when Bose atoms are in the normal state, the density profile exhibits Friedel oscillations. The second main finding is that when Bose atoms are in the BEC state, the density profile exhibits a sharp peak with extremely narrow width. The third main finding is that the central peak of the spatial density is a monotonically increasing function of the number of atoms N but is a monotonically decreasing function of temperature T.