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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]

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]

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]

In this paper, we solve the question completely raised by Feng Qi and Dongkyu Lim in paper "Monotonicity properties for a ratio of finite many gamma functions" published in Advances in Difference Equations and get some properties about ratios of finitely many gamma functions such as complete monotonicity, logarithmically complete monotonicity, the Bernstein function property, null point and extreme value. [ABSTRACT FROM AUTHOR]

In this paper, a new approximation formula for the gamma function and some of its symmetric inequalities are established. We prove the superiority of our results over Yang and Tian's approximation formula for the gamma function of order v − 9 . [ABSTRACT FROM AUTHOR]

In this note, we prove that x n 1 - e - x (n) > 0 for all x ∈ (log 2 , ∞) and n ∈ N . This result improves a theorem of Al-Musallam and Bustoz (Ramanujan J 11:399–402, 2006). [ABSTRACT FROM AUTHOR]

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]

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]

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]

We derive explicit closed-form formulas for the standard Clausen functions C l 2 m (x) and C l 2 m − 1 (x) in terms of the Hurwitz zeta function, where m is a positive integer. [ABSTRACT FROM AUTHOR]

Abstract The normalization of the generalized Bessel functions U σ , r $\mathrm{U}_{\sigma,r}$ ( σ , r ∈ C ) $(\sigma,r\in \mathbb{C}\mathbbm{)}$ defined by U σ , r ( z ) = z + ∑ j = 1 ∞ ( − r ) j 4 j ( 1 ) j ( σ ) j z j + 1 $$\begin{aligned} \mathrm{U}_{\sigma,r}(z)=z+\sum_{j=1}^{\infty} \frac{(-r)^{j}}{4^{j} (1)_{j}(\sigma )_{j}}z^{j+1} \end{aligned}$$ was introduced, and some of its geometric properties have been presented previously. The main purpose of the present paper is to complete the results given in the literature by employing a new procedure. We first used an identity for the logarithmic of the gamma function as well as an inequality for the digamma function to establish sufficient conditions on the parameters so that U σ , r $\mathrm{U}_{\sigma,r}$ is starlike or convex of order α ( 0 ≤ α ≤ 1 ) $(0\leq \alpha \leq 1)$ in the open unit disk. Moreover, the starlikeness and convexity of U σ , r $\mathrm{U}_{\sigma,r}$ have been considered where the leading concept of the proofs comes from the starlikeness of the power series f ( z ) = ∑ j = 1 ∞ A j z j $f(z)=\sum_{j=1}^{\infty}A_{j}z^{j}$ and the classical Alexander theorem between the classes of starlike and convex functions. We gave a simple proof to show that our conditions are not contradictory. Ultimately, the close-to-convexity of ( z cos z ) ∗ U σ , r $(z\cos \sqrt{z} ) \ast \mathrm{U}_{\sigma,r}$ and ( sin z ) ∗ U σ , r ( z 2 ) z $(\sin z ) \ast \frac {\mathrm{U}_{\sigma,r}(z^{2})}{z}$ have been determined, where “∗” stands for the convolution between the power series.

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]

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]

In this paper, we present certain class of strictly positive definite functions related to the extended Hurwitz-Lerch Zeta function, digamma function and the modified Bessel function of first kind. The key tools in our proofs are some explicit formulas for the modified Bessel function of the first kind and a new integral representation of the extended Hurwitz-Lerch Zeta function when their terms contain the Fox H-function. As applications, monotonicity properties and some new functional inequalities of the aforementioned functions are established. [ABSTRACT FROM AUTHOR]

Let ψ be the digamma function and let L(a,b) = (b − a)/log(b/a) be the logarithmic mean of a and b. We prove that the inequality holds for all real numbers a and b with b > a ≥ α0. Here, α0 ≈ 0.56155 is the only positive solution of The constant lower bound α0 is best possible. This refines a result of Chu, Zhang and Tang, who showed that (*) is valid for b > a ≥ 2. Moreover, we prove that the following companion to (*) holds for all a and b with b > a > 0, [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, a new approximation formula for the gamma function and some of its symmetric inequalities are established. We prove the superiority of our results over Yang and Tian’s approximation formula for the gamma function of order v−9.

The normalization of the generalized Bessel functions U σ , r (σ , r ∈ C) defined by U σ , r (z) = z + ∑ j = 1 ∞ (− r) j 4 j (1) j (σ) j z j + 1 was introduced, and some of its geometric properties have been presented previously. The main purpose of the present paper is to complete the results given in the literature by employing a new procedure. We first used an identity for the logarithmic of the gamma function as well as an inequality for the digamma function to establish sufficient conditions on the parameters so that U σ , r is starlike or convex of order α (0 ≤ α ≤ 1) in the open unit disk. Moreover, the starlikeness and convexity of U σ , r have been considered where the leading concept of the proofs comes from the starlikeness of the power series f (z) = ∑ j = 1 ∞ A j z j and the classical Alexander theorem between the classes of starlike and convex functions. We gave a simple proof to show that our conditions are not contradictory. Ultimately, the close-to-convexity of (z cos z) ∗ U σ , r and (sin z) ∗ U σ , r (z 2) z have been determined, where "∗" stands for the convolution between the power series. [ABSTRACT FROM AUTHOR]

We define a surface that interpolates the ballot numbers in the Catalan triangle corresponding to every pair of nonnegative integers (except for the origin). We study the geometric properties of this surface and prove that it contains exactly five half-lines. The mean curvature and the Gauss curvature of the surface are also calculated. [ABSTRACT FROM AUTHOR]

This paper deals with the parameters of a natural extension of the Wishart distribution, that is the Riesz distribution on the space of symmetric matrices. We estimate the shape parameter using two different approaches. The first one is based on the method of moments, we give its expression and investigate some of its properties. The second represents the maximum likelihood estimator. Unfortunately, in this case we do not have an explicit formula for this estimator. This latter is expressed in terms of the digamma function and sample mean of log-gamma variables. However, we derive the strong consistency and asymptotic normality properties of this estimator. A numerical comparative study between the two estimators is carried out in order to test the performance of the proposed approaches. For the second parameter, that is the scale parameter, we prove that the distribution of the maximum likelihood estimator given by Kammoun et al. (J Statist Prob Lett 126:127–131, 2017) is related to the Riesz distribution. We examine some properties concerning this estimator and we assess its performance by a numerical study. [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]

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]

In this paper, we establish new generalizations of the Hermite-Hadamard-type inequalities. These inequalities are formulated in terms of modules of certain powers of proper functions. Generalizations for convex functions are also considered. As applications, some new inequalities for the digamma function in terms of the trigamma function and some inequalities involving special means of real numbers are given. The results also include estimates via arithmetic, geometric and logarithmic means. The examples are derived in order to demonstrate that some of our results in this paper are more exact than the existing ones and some improve several known results available in the literature. The constants in the derived inequalities are calculated; some of these constants are sharp. As a visual example, graphs of some technically important functions are included in the text. [ABSTRACT FROM AUTHOR]

We present a summation rule using Mellin's transform to give short proofs of some important classical relations between special functions and Bernoulli and Euler polynomials. For example, the values of the Hurwitz zeta function at the negative integers are expressed in terms of Bernoulli polynomials. We also show identities involving exponential and Hermite polynomials. [ABSTRACT FROM AUTHOR]

The arithmetic nature of the Euler's constant γ is one of the biggest unsolved problems in number theory from almost three centuries. In an attempt to give a partial answer to the arithmetic nature of γ , Murty and Saradha made a conjecture on linear independence of digamma values. In particular, they conjectured that for any positive integer q > 1 and a field K over which the q-th cyclotomic polynomial is irreducible, the digamma values namely ψ (a / q) where 1 ≤ a ≤ q with (a , q) = 1 are linearly independent over K. Further, they established a connection between the arithmetic nature of the Euler's constant γ to the above conjecture. In this article, we first prove that the conjecture is true with at most one exceptional q. Later on we also make some remarks on the linear independence of these digamma values with the arithmetic nature of the Euler's constant γ . [ABSTRACT FROM AUTHOR]

In Bayesian analysis, empirical and hierarchical methods are two main approaches for the estimation of the parameter(s) involved in the prior distribution of one parameter. But in the multi-parameter model, e.g., Gamma(α; p), where both the parameters are unknown, idea of the 'Partial Bayes (PB) Estimation' is introduced. When we do not have proper belief regarding the joint parameters of the distribution of the variable and when we are estimating one parameter in presence of others, such method may be used. Partial Bayes estimation of the scale parameter p is done by putting the estimate of the another parameter α obtained by some other classical method in case of two parameter Gamma distribution. Using non-informative prior and computing the risk, it is found that the Partial Bayes estimator has less risk than the Bayes estimator. For this, simulation studies for some choices of shape parameter values have been done. In case of the shape parameter, posterior mean and posterior variance are evaluated through simulations to obtain the risk values for estimator of α with known scale parameter. Finally after fitting this distribution, two real datasets are illustrated to see the performance of the Partial Bayes estimator. [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]

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.

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]

In this paper, several novel inequalities are examined for the product of two s-convex functions in the fourth sense. Also, some applications regarding special means and digamma functions are presented. [ABSTRACT FROM AUTHOR]

We prove a short general theorem which immediately implies some classical results of Hasse, Guillera and Sondow, Paolo Amore, and also Alzer and Richards. At the end we obtain a new representation for Euler's constant γ. The theorem transforms every Taylor series into a series depending on a parameter. [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 3F2 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.

Motivated by several conjectures posed in the paper "F. Qi and A.-Q. Liu, Completely monotonic degrees for a difference between the logarithmic and psi functions, J. Comput. Appl. Math. 361 (2019), 366-371; https://doi.org/10.1016/j.cam.2019.05.001", the authors bound several completely monotonic degrees of the remainders in the asymptotic expansions of the logarithm of the gamma function and in the asymptotic expansions of the logarithm of the digamma function. [ABSTRACT FROM AUTHOR]

Abstract Several conjectures posed by Qi on completely monotonic degrees of remainders for the asymptotic formulas of the digamma and trigamma functions are proved.

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.

In this paper, we establish new generalizations of the Hermite-Hadamard-type inequalities. These inequalities are formulated in terms of modules of certain powers of proper functions. Generalizations for convex functions are also considered. As applications, some new inequalities for the digamma function in terms of the trigamma function and some inequalities involving special means of real numbers are given. The results also include estimates via arithmetic, geometric and logarithmic means. The examples are derived in order to demonstrate that some of our results in this paper are more exact than the existing ones and some improve several known results available in the literature. The constants in the derived inequalities are calculated; some of these constants are sharp. As a visual example, graphs of some technically important functions are included in the text.

We investigate conditions for logarithmic complete monotonicity of product ratios of gamma and q-gamma functions whose arguments are linear functions of the variable. We give necessary and sufficient conditions in terms of nonnegativity of a certain explicitly written measure in the q case and of a certain elementary function in the classical q = 1 case. In the latter case we further provide simple new sufficient conditions leading to many new examples of logarithmically completely monotonic gamma ratios. Finally, we apply some of our results to study monotonicity of some gamma ratios and rational functions. [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]

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]

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]

This contribution is a follow-up to author’s previous published papers and deals with the sum of the series of reciprocals of the cubic polynomials with one zero and double non-zero integer root. We derive the formula for the sum of these series and verify it by some examples using the basic programming language of the computer algebra system Maple 16. This contribution can be an inspiration for teachers who are teaching the topic Infinite series or as a subject matter for work with talented students.

Abstract Let Hn=∑r=1n1/r $H_{n} = \sum_{r=1}^{n} 1/r$ and Hn(x)=∑r=1n1/(r+x) $H_{n}(x) = \sum_{r=1}^{n} 1/(r+x)$. Let ψ(x) $\psi(x)$ denote the digamma function. It is shown that Hn(x)+ψ(x+1) $H_{n}(x) + \psi(x+1)$ is approximated by 12logf(n+x) $\frac{1}{2}\log f(n+x)$, where f(x)=x2+x+13 $f(x) = x^{2} + x + \frac{1}{3}$, with error term of order (n+x)−5 $(n+x)^{-5}$. The cases x=0 $x = 0$ and n=0 $n = 0$ equate to estimates for Hn−γ $H_{n} - \gamma $ and ψ(x+1) $\psi(x+1)$ itself. The result is applied to determine exact bounds for a remainder term occurring in the Dirichlet divisor problem.

Abstract Let Γ(x) $\varGamma (x)$ denote the classical Euler gamma function. The logarithmic derivative ψ(x)=[lnΓ(x)]′=Γ′(x)Γ(x) $\psi (x)=[\ln \varGamma (x)]'=\frac{\varGamma '(x)}{ \varGamma (x)}$, ψ′(x) $\psi '(x)$, and ψ″(x) $\psi ''(x)$ are, respectively, called the digamma, trigamma, and tetragamma functions. In the paper, the authors survey some results related to the function [ψ′(x)]2+ψ″(x) $[\psi '(x)]^{2}+ \psi ''(x)$, its q-analogs, its variants, its divided difference forms, several ratios of gamma functions, and so on. These results include the origins, positivity, inequalities, generalizations, completely monotonic degrees, (logarithmically) complete monotonicity, necessary and sufficient conditions, equivalences to inequalities for sums, applications, and the like. Finally, the authors list several remarks and pose several open problems.

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 Γμ(v) and study the completely monotonic (CM) property of a function involving Γμ(v) and the generalized digamma function ψμ(v). As a consequence, we establish some bounds for Γμ(v), ψμ(v) and polygamma functions ψμ(r)(v), r≥1.

Abstract 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.

This paper deals with the study of the reliability of multicomponent stress–strength model assuming that the strength components are independently and identically distributed as power transformed half-logistic (PHL) distribution. The strength components which are subject to a common stress are assumed to be independent with either Weibull distribution or PHL distribution. The maximum likelihood estimates of the multicomponent stress–strength reliability and its asymptotic confidence interval under the above said conditions are obtained. The Bayes estimates of the multicomponent stress–strength reliability are also obtained under squared error loss function and using gamma priors for the parameters. To evaluate the performance of the procedure, a simulation study is considered. For illustration purpose of the proposed model, two real life examples are given. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present closed forms for various types of infinite series involving Fibonacci (Lucas) numbers and the Riemann zeta function at integer arguments. To prove our results, we will apply some conventional arguments and combine the Binet formulas for these sequences with generating functions involving the Riemann zeta function and some known series evaluations. Among the results derived in this paper, we will establish that *** where is the familiar Euler-Mascheroni constant. [ABSTRACT FROM AUTHOR]