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We presented some monotonicity properties for the k-generalized digamma function k(h) and we established some new bounds for (s) k (h), s N {0}, which refine recent results. [ABSTRACT FROM AUTHOR]

We calculate some infinite sums containing the digamma function in closed form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta incomplete function, reduction formulas of 3 F 2 hypergeometric functions, or a definite integral which does not seem to be tabulated in the most common literature. As an application of certain sums involving the digamma function, we calculated some reduction formulas for the parameter differentiation of the Mittag–Leffler function and the Wright function. [ABSTRACT FROM AUTHOR]

Numerous series representations for various special functions and mathematical constants have been developed by many authors. The aim of this article is to establish two parameterized series representations for the digamma function that seem interesting due to their independence from the given parameters. Among many particular cases of our two main findings, some are covered in the examples. [ABSTRACT FROM AUTHOR]

We calculate some finite and infinite sums containing the digamma function in closed form. For this purpose, we differentiate selected reduction formulas of the hypergeometric function with respect to the parameters applying some derivative formulas of the Pochhammer symbol. Additionally, we compare two different differentiation formulas of the generalized hypergeometric function with respect to the parameters. For some particular cases, we recover some results found in the literature. Finally, all the results have been numerically checked. [ABSTRACT FROM AUTHOR]

We present functional inequalities for the q-digamma function ψ q = Γ ′ q / Γ q , where Γ q denotes the q-gamma function. Among others, we prove that for all positive real numbers a, b with a ≠ b we have ψ q (a) + ψ q (b) 2 < ψ q ( ab ) , 0 < q < 1 , and a ψ q (e 1 / a ) + b ψ q (e 1 / b ) < (a + b) ψ q (e 2 / (a + b) ) , 0 < q < 1. [ABSTRACT FROM AUTHOR]

We consider some properties of integrals considered by Hardy and Koshliakov that have connections to the digamma function. We establish a new general integral formula that provides a connection to the polygamma function. We also obtain lower and upper bounds for Hardy's integral through properties of the digamma function. [ABSTRACT FROM AUTHOR]

A fast algorithm for calculating the logarithmic derivative of the Euler gamma function based on the BVE method is constructed. The complexity of the algorithm is close to optimal. The structure of the algorithm allows its parallelization. [ABSTRACT FROM AUTHOR]

We consider some properties of integrals considered by Hardy and Koshliakov that have connections to the digamma function. We establish a new general integral formula that provides a connection to the polygamma function. We also obtain lower and upper bounds for Hardy's integral through properties of the digamma function. [ABSTRACT FROM AUTHOR]

Eisenstein derived addition formulas for the Weierstrass zeta function from the addition formula for the cotangent function and the fact that the Weierstrass zeta function can be represented as an infinite sum of the cotangent functions. In this paper, we apply this idea of Eisenstein's to the addition type formula for the double cotangent function, established by the author. We show that the elliptic digamma function, defined by the logarithmic derivative of the elliptic gamma function, satisfies an addition type formula. This formula includes the addition formula for the Weierstrass zeta function, evaluation formulas for the double Eisenstein series introduced by Tsumura and the double shuffle relations for the double Eisenstein series, proved by Gangl-Kaneko-Zagier. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions Γ q (x + 1) / Γ q (x + s) for all real number s and 0 < q ≠ 1 in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all 0 < s < 1 when letting q → 1 and a new inequality for all s > 1 . [ABSTRACT FROM AUTHOR]

In this paper, the authors establish some asymptotic formulas and two-sided inequalities for the ratio of the gamma function in terms of the digamma function. Considering the application of the gamma function to central Bernoulli coefficients, the authors give some better approximations and two-sided inequalities of central Bernoulli coefficients. Finally, for demonstrating the superiority of our results, some numerical computations are given. [ABSTRACT FROM AUTHOR]

Motivated by the substantial development in the theory of digamma function, we derive some new identities for the digamma function. These new identities enable us to compute the values of the digamma function for fractional orders in an analogous manner. Also, we mention two errata, found in Jensen's article (An elementary exposition of the theory of the Gamma function, 1916), and present their correct forms. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The 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 thenth powers of the reciprocals of its zeros, for. We make a parallel study of the zeros of the logarithmic derivative of the BarnesG-function. We also compare asymptotic estimates of the zeros of the digamma function and those of its BarnesG-function analogue. [ABSTRACT FROM PUBLISHER]

Abstract: We extend some known inequalities for the digamma function to the -digamma function. Furthermore, we estimate the unique positive real root of the -digamma function . [Copyright &y& Elsevier]

In the paper, the authors establish three kinds of inequalities for the trigamma function in terms of the exponential function to powers of the digamma function. These newly established inequalities extend some known results. The method in the paper utilizes some facts from the asymptotic theory and is a natural way to solve problems for approximating some quantities for large values of the variable. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to refine some recent results stated by Alzer and Grinshpan [H. Alzer, A.Z. Grinshpan, Inequalities for the gamma and -gamma functions J. Approx. Theory 144 (2007) 67–83] and Batir [N. Batir, An interesting double inequality for Euler’s gamma function J. Inequal. Pure Appl. Math. 5(4) (2004) article 97] concerning the problem of approximation of the gamma function in terms of the digamma function. [Copyright &y& Elsevier]

Abstract: The goal of this paper is to prove the following asymptotic formula where is the Euler Gamma function and is the digamma function, namely, the logarithmic derivative of . Moreover, optimal values of parameters are calculated in such a way that this asymptotic convergence is the best possible. [Copyright &y& Elsevier]

Abstract: Let denote the digamma function. We study the linear independence of at rational arguments over algebraic number fields. We also formulate a variant of a conjecture of Rohrlich concerning linear independence of the log gamma function at rational arguments and report on some progress. We relate these conjectures to non-vanishing of certain L-series. [Copyright &y& Elsevier]

Abstract: Let denote the digamma function, that is, the logarithmic derivative of Euler''s Γ-function. Let q be a positive integer greater than 1 and γ denote Euler''s constant. We show that all the numbers are transcendental. We also prove that at most one of the numbers is algebraic. [Copyright &y& Elsevier]

Abstract: The aim of this work is to obtain the so-called standard lemmas on irrationality bases using the principles of Chudnovsky and then apply them to obtain conditional irrationality measures for values of the digamma function. [Copyright &y& Elsevier]

this paper, we present the (p, q)-analogues of some inequalities concerning the digamma function. Our results generalize some earlier results. [ABSTRACT FROM AUTHOR]

An open problem about integral representation of , proposed recently by Pain (2023), is resolved by integration by parts. More general integrals are examined by manipulating the beta integral and digamma function. [ABSTRACT FROM AUTHOR]

We explicitly construct rational approximations to the numbers ln( b) − ψ( a + 1), where ψ is the logarithmic derivative of the Euler gamma function. We prove formulas expressing the numerators and the denominators of the approximations in terms of hypergeometric sums. This generalizes the Aptekarev construction of rational approximations for the Euler constant γ. As a consequence, we obtain rational approximations for the numbers π/2 ± γ. The proposed construction is compared with rational Rivoal approximations for the numbers γ + ln( b). We verify assumptions put forward by Rivoal on the denominators of rational approximations to the numbers γ + ln( b) and on the general denominators of simultaneous approximations to the numbers γ and ζ(2) − γ. [ABSTRACT FROM AUTHOR]

Let k , t be coprime integers, and let 1 ≤ r ≤ t. We let D k × (r , t ; n) denote the total number of parts among all k -indivisible partitions (i.e., those partitions where no part is divisible by k) of n which are congruent to r modulo t. In previous work of the authors [3] , an asymptotic estimate for D k × (r , t ; n) was shown to exhibit unpredictable biases between congruence classes. In the present paper, we confirm our earlier conjecture in [3] that there are no "ties" (i.e., equalities) in this asymptotic for different congruence classes. To obtain this result, we reframe this question in terms of L -functions, and we then employ a nonvanishing result due to Baker, Birch, and Wirsing [1] to conclude that there is always a bias towards one congruence class or another modulo t among all parts in k -indivisible partitions of n as n becomes large. [ABSTRACT FROM AUTHOR]

We present some properties associated with the monotonicity and the complete monotonicity of the psi (or digamma) function and establish the higher-order estimate for the familiar Euler–Mascheroni constant γ. Relevant connections of the results presented here with those derived in earlier works are also pointed out. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this work is to find “good” approximations to the Digamma function . We construct an infinite family of “basic” functions covering the Digamma function. These functions are shown to approximate locally and asymptotically, and it is shown that for any , there exists an such that . Local and global bounding error functions are found and, as a consequence, new inequalities for the Digamma function are introduced. The approximations are compared to another, well-known, approximation of the Digamma function and we show that an infinite number of members of the family are better. [Copyright &y& Elsevier]

The aim of this paper is to establish new sharp upper and lower bounds for the gamma and digamma functions, starting from the Stirling's formula. [ABSTRACT FROM AUTHOR]

Let H n = ∑ r = 1 n 1 / r and H n (x) = ∑ r = 1 n 1 / (r + x) . Let ψ (x) denote the digamma function. It is shown that H n (x) + ψ (x + 1) is approximated by 1 2 log f (n + x) , where f (x) = x 2 + x + 1 3 , with error term of order (n + x) − 5 . The cases x = 0 and n = 0 equate to estimates for H n − γ and ψ (x + 1) itself. The result is applied to determine exact bounds for a remainder term occurring in the Dirichlet divisor problem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For a fixed prime p, Murty and Saradha (Acta Arith 133:349–362, 2008) studied the transcendental nature of special values of the p-adic digamma function, denoted as ψ p (r / p) + γ p . This research was later extended by Chatterjee and Gun in 2014, who investigated the case of ψ p (r / p n) + γ p , for any integer n > 1 . In this article, we generalize their results for distinct prime powers and explore the transcendental nature of the p-adic digamma values, with at most one exception. Further, we investigate the multiplicative independence of cyclotomic numbers satisfying certain conditions. Using this, we prove the transcendental nature of p-adic digamma values corresponding to ψ p (r / p q) + γ p , where p, q are distinct primes. [ABSTRACT FROM AUTHOR]

In this paper, we detail closed form estimators for beta distribution that are simpler than those proposed by Tamae, Irie and Kubokawa. The proposed estimators are shown to have smaller asymptotic variances and smaller asymptotic covariances compared to Tamae estimators and maximum likelihood estimators. The proposed estimators are also shown to perform better in real data applications. [ABSTRACT FROM AUTHOR]

Nawa and Nadarajah [Statistics, 2023, doi: ] proposed closed form estimators for a bivariate gamma distribution. But these estimators do not have a direct multivariate extension. In this paper, we propose alternative closed form estimators with direct extensions to the trivariate and multivariate cases. These estimators can have smaller asymptotic variances and smaller asymptotic covariances compared to Zhao et al.'s estimators, method of moments estimators and Nawa and Nadarajah's estimators. [ABSTRACT FROM AUTHOR]

Motivated by Zhao, Jang and Kim [Journal of Multivariate Analysis, 191, 2022, article number 105109], we propose new closed form estimators for a bivariate gamma distribution. The new estimators are simpler. In addition, they can have smaller asymptotic variances and smaller asymptotic covariances compared to Zhao et al.'s estimators and method of moments estimators. The new estimators can also perform better in real-data applications. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate that Δn(x) = Γ(n-1) (x) Γ(n+1) (x) - [Γ(n) (x)]² are strictly convex on (0;+∞) for all n = 1, 3, . . ., and there exists a unique global minimum xn of Δn (x) for each odd positive integer n. On the other hand, we develop the algorithm for calculating the global minimum or the zero point of Δn (x) based on the Newton's method and the recurrence relation of Γ(n) (x) associated with the Digamma function, the numerical results show that it's more effective than other two algorithms. Furthermore, we find that Δn (xn) is strictly increasing and Δn (xn) » α = 0:6359 . . . for all odd integers n ≥ 7 through the numerical experiments. [ABSTRACT FROM AUTHOR]

Recently, interesting novel summation formulae for hypergeometric-type series containing a digamma function as a factor have been established by Miller (2006 J. Phys. A: Math. Gen. 39 3011-20) mainly by exploiting already known results and using certain transformation and reduction formulae in the theory of the Kampé de Fériet double hypergeometric function. It is shown here that these, and several other series of this type, can be closed-form summed by simpler and more direct arguments based only on a derivative formula for the Pochhammer symbol and the theory of the digamma (or ps) function and generalized hypergeometric function. In addition, a new reduction formula for the Kampé de Fériet function F^{\scriptstyle\, {\rm{1:2;1}} \hfill \atop \scriptstyle\, {\rm{1: 1; 0}} \hfill}[z,z] is obtained. [ABSTRACT FROM AUTHOR]

For integers k , t ≥ 2 , and 1 ≤ r ≤ t let D k × (r , t ; n) be the number of parts among all k -indivisible partitions of n (i.e., partitions where all parts are not divisible by k) of n that are congruent to r modulo t. Using Wright's circle method, we derive an asymptotic for D k × (r , t ; n) as n → ∞ when k , t are coprime. The main term of this asymptotic does not depend on r , and so, in a weak asymptotic sense, the parts are equidistributed among congruence classes. However, inspection of the lower order terms indicates a bias towards different congruence classes modulo t. This induces an ordering on the congruence classes modulo t , which we call the k -indivisible ordering. We prove that for k ≥ 6 (t 2 − 1) π 2 the k -indivisible ordering matches the natural ordering. We also explore the properties of these orderings when k < 6 (t 2 − 1) π 2 . • Biases among congruence classes for parts in k -indivisible partitions. • The Circle Method in the study of integer partitions. • Connections between L -functions and parts in integer partitions. • Biases understood via linear combinations of the digamma function. [ABSTRACT FROM AUTHOR]

A famous identity of Gauss gives a closed form expression for the values of the digamma function Ψ (x) at rational arguments x in terms of elementary functions. Linear combinations of such values are intimately connected with a conjecture of Erdős which asserts non vanishing of an infinite series associated to a certain class of periodic arithmetic functions. In this note we give a different proof for the identity of Gauss using an orthogonality like relation satisfied by these functions. As a by product we are able to give a new interpretation for nth Catalan number in terms of these functions. [ABSTRACT FROM AUTHOR]

We propose closed form estimators for the most popular bivariate beta distribution known to date and derive their asymptotic variances. The asymptotic properties of the proposed estimators are compared to asymptotic properties of the maximum likelihood estimators. The proposed estimators can have smaller asymptotic variances and smaller asymptotic covariances compared to maximum likelihood estimators. [ABSTRACT FROM AUTHOR]

In the paper, the authors bound the gamma function, the digamma function, and the harmonic numbers in terms of the exponent of fractional functions and several trigonometric functions such as the tangent, hyperbolic tangent, secant, and cosecant functions. [ABSTRACT FROM AUTHOR]

In the paper, the authors bound the gamma function, the digamma function, and the harmonic numbers in terms of the exponent of fractional functions and several trigonometric functions such as the tangent, hyperbolic tangent, secant, and cosecant functions. [ABSTRACT FROM AUTHOR]

The symmetric patterns that inequalities contain are reflected in researchers' studies in many mathematical sciences. In this paper, we prove an asymptotic expansion for the generalized gamma function Γ μ (v) and study the completely monotonic (CM) property of a function involving Γ μ (v) and the generalized digamma function ψ μ (v) . As a consequence, we establish some bounds for Γ μ (v) , ψ μ (v) and polygamma functions ψ μ (r) (v) , r ≥ 1 . [ABSTRACT FROM AUTHOR]

We evaluate some finite and infinite sums involving q -trigonometric and q -digamma functions. Upon letting q approach 1 , one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a theta product formula of Jacobi and Gosper's q -trigonometric identities. [ABSTRACT FROM AUTHOR]

In this paper, we have developed the generalized expression ... and some formulae of this generalized formula. [ABSTRACT FROM AUTHOR]

We present our computationally efficient approach to modeling tunable graphene-based active metadevices, where the integral multivariate surface conductivity is reformulated in the time and frequency domains with physically interpretable and fast-to-compute integration-free terms. The derivation is built on an expansion to power series of $ {z}= -\text {exp}(-\mu /k_{B}T)$ that converges very fast for non-zero chemical potential values. The model reveals interesting decomposition of graphene response into a universal constant term plus a damped oscillator (digamma functions in the frequency domain) plus non-oscillating correction terms for near-zero potentials. The number of terms in that series is analyzed theoretically for a given accuracy. In practice, only a few series terms are required, making our approach very efficient for simulation of active metasurfaces compared with direct integration of Kubo’s formulas. A simple performance test comparing run times with our code versus the numerical integration of the original Kubo’s formulas demonstrates a speedup exceeding 103. The proposed models can be critical for the initial ellipsometric characterization of graphene and advanced global optimization of graphene-controlled metadevices. [ABSTRACT FROM AUTHOR]

Recently, it was conjectured that the first generalized Stieltjes constant at rational argument may be always expressed by means of Euler's constant, the first Stieltjes constant, the Γ-function at rational argument(s) and some relatively simple, perhaps even elementary, function. This conjecture was based on the evaluation of γ 1 ( 1 / 2 ) , γ 1 ( 1 / 3 ) , γ 1 ( 2 / 3 ) , γ 1 ( 1 / 4 ) , γ 1 ( 3 / 4 ) , γ 1 ( 1 / 6 ) , γ 1 ( 5 / 6 ) , which could be expressed in this way. This article completes this previous study and provides an elegant theorem which allows to evaluate the first generalized Stieltjes constant at any rational argument. Several related summation formulae involving the first generalized Stieltjes constant and the Digamma function are also presented. In passing, an interesting integral representation for the logarithm of the Γ-function at rational argument is also obtained. Finally, it is shown that similar theorems may be derived for higher Stieltjes constants as well; in particular, for the second Stieltjes constant the theorem is provided in an explicit form. [ABSTRACT FROM AUTHOR]

We offer two new Mellin transform evaluations for the Riemann zeta function in the region 0 < ℜ (s) < 1. Some discussion is offered in the way of evaluating some further Fourier integrals involving the Riemann xi function. [ABSTRACT FROM AUTHOR]

In this paper we investigate some properties of the Hyperharmonic function defined by ... Using this definition we introduce harmonic numbers with complex index and we give some series of these numbers. Also formulas for the calculation of harmonic numbers with rational index are obtained. For the simplicity of differentiation we reorganized representation of H(w) z. With the help of this new form we get higher order derivatives of the Hyperharmonic function more easily. Besides these, owing to the fact that the Hyperharmonic function is composed of some important functions, we interested in properties and connections of it. We get connections between the Hyperharmonic function and trigonometric functions. Infinite product representation, integral representation and differentiation identities of this function are also obtained. [ABSTRACT FROM AUTHOR]