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Let ψ (x) denote the digamma function, that is, the logarithmic derivative of the classical Euler gamma function Γ (x) . In the paper, the authors discover the monotonic properties of the functions ψ (n) (x) x ψ (n + 1) (x) and ψ (n) (x) ψ (n) (1 x) for n ≥ 0 on (0 , ∞) . With the aid of these monotonic properties, the authors confirm the positivity of the function ψ (x) + x ψ ′ (x) − ψ (x) ψ (1 x) on (0 , ∞) . The authors also pose five open problems, generalizing the latter two of the three functions mentioned above and their related conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate one in the family of four parametric linear Euler sums as given by Alzer and Choi and later extended by Sofo and Choi. We give explicit expressions for a variety of special cases involving a free parameter. Our analysis incorporates the properties of the polylogarithm function, commonly referred to as the Fermi–Dirac integral. A reciprocity property is utilized to highlight an explicit representation of a particular skew harmonic linear Euler sum. A number of examples are also given which highlight the theorems. This work generalizes some results in the published literature. [ABSTRACT FROM AUTHOR]

Definite integrals containing gamma functions and other special functions are obtained. [ABSTRACT FROM AUTHOR]

We aim to investigate the four types of variant Euler harmonic sums. Also, as corollaries, we provide particular examples of our core findings, some of whose further instances are evaluated in terms of basic and well-known functions as well as certain mathematical constants. We explore relevant linkages between our results and those of other previously established studies. An examination of a specific case of one result shows a relationship to series involving zeta functions, which is also a popular area of research. [ABSTRACT FROM AUTHOR]

We investigate the linear arrangement of Euler harmonic sums that may be expressed in closed form in terms of special functions such as the classical Riemann zeta function and the Dirichlet eta function. Particular emphasis is given to Euler harmonic sums with even weight. New examples highlighting the theorems will be presented. [ABSTRACT FROM AUTHOR]

We offer intriguing new insights into parameterized finite binomial sums, revealing elegant identities such as ∑ k = 0 , k ≠ n m + n (− 1) k n − k m + n k = (− 1) n m + n n (H m − H n) , where n , m are non-negative integers and H n is the harmonic number. These formulas beautifully capture the intricate relationship between harmonic numbers and binomial coefficients, providing a fresh and captivating perspective on combinatorial sums. [ABSTRACT FROM AUTHOR]

In this study we investigate a family of skew linear harmonic Euler sums involving some free parameters. Our analysis involves using the properties of the polylogarithm function, commonly referred to as the Bose-Einstein integral. A reciprocity property is utilized to highlight an explicit representation for a particular skew harmonic linear Euler sum. A number of examples are also given which highlight the theorems. This work generalizes some results in the published literature and introduces some new results.

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. [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]

We consider an inequality concerning absolute values of the Riemann zeta function at places symmetric with respect to the critical line. The study concludes the investigations started by Spira and complements Trudgian's results. [ABSTRACT FROM AUTHOR]

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector X=(X1,…,Xd){\boldsymbol{X}}=\left({X}_{1},\ldots ,{X}_{d}) of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on X{\boldsymbol{X}} is shown to be strictly weaker than the assumption that the density of the random vector (∣X1∣,…,∣Xd∣)\left(| {X}_{1}| ,\ldots ,| {X}_{d}| ) is multivariate totally positive of order 2, abbreviated MTP2{\text{MTP}}_{2}, for which the GPI is already known to hold. Under this condition, the paper highlights a new link between the GPI and the monotonicity of a certain ratio of gamma functions.

We prove two conjectures due to Sun concerning binomial-harmonic sums. First, we introduce a proof of a formula for Catalan’s constant that had been conjectured by Sun in 2014. Then, using a similar approach as in our first proof, we solve an open problem due to Sun involving the sequence of alternating odd harmonic numbers. Our methods, more broadly, allow us to reduce difficult binomial-harmonic sums to finite combinations of dilogarithms that are evaluable using previously known algorithms. [ABSTRACT FROM AUTHOR]

In the paper, the authors present a brief overview and survey of the scientific work by Chinese mathematician Feng Qi and his coauthors. [ABSTRACT FROM AUTHOR]

n this paper, the complete monotonicity of 1/( arctan 𝑥) is proved. This problem was posted by F. Qi and R. P. Agarwal as the eighth open problem of collection of eight open problems.

In the paper, the authors introduce a matrix-parametrized generalization of the multinomial probability mass function that involves a ratio of several multivariate gamma functions. They show the logarithmically complete monotonicity of this generalization and derive several inequalities involving ratios of multivariate gamma functions. [ABSTRACT FROM AUTHOR]

Closed-form expressions for the three general linear Euler-type sums containing the harmonic numbers Hn and the skew-harmonic numbers Hn *** where q is a positive integer greater than or equal to zero or two as needed to ensure convergence are given. Closed-form expressions for several other closely related generalised logarithmic integrals and sums are also presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the paper, the author presents bounds for completely monotonic degree of a remainder for an asymptotic expansion of the trigamma function. This result partially confirms one in a series of conjectures on completely monotonic degrees of remainders of asymptotic expansions for the logarithm of the gamma function and for polygamma functions.

We present expressions that relate the Riemann Zeta function, the Polygamma function in the case 1/4 as well as the Clausen functions for π/2, having been obtained by means of elementary procedures from the function cos(nx)/nm-1. [ABSTRACT FROM AUTHOR]

Abstract We prove a monotonicity property of the Hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. We provide a probabilistic interpretation of our result by exploiting a connection between Hurwitz zeta function and the cumulants of the beta-exponential distribution.

We point out that the proof of Lemma 5 in the article "A harmonic mean inequality for the polygamma function" by S. Das and A. Swaminathan contains a small mistake. Giving a new and proper proof of Lemma 5 is the main aim of this paper. [ABSTRACT FROM AUTHOR]

In the paper, the author presents bounds for completely monotonic degree of a remainder for an asymptotic expansion of the trigamma function. This result partially confirms one in a series of conjectures on completely monotonic degrees of remainders of asymptotic expansions for the logarithm of the gamma function and for polygamma functions. [ABSTRACT FROM AUTHOR]

In the paper, by virtue of convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the Fisher–Rao manifold of beta distributions. [ABSTRACT FROM AUTHOR]

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.

In this paper, the authors prove some inequalities and completely monotonic properties of polygamma functions. As an application, we give lower bound for the zeta function on natural numbers. Partially, we answer the fifth and sixth open problems listed in [F. Qi and R. P. Agarwal, On complete monotonicity for several classes of functions related to ratios of gamma functions, J. Inequalities Appl.2019(36) (2019) 42]. We propose two open problems on completely monotonic functions related to polygamma functions. [ABSTRACT FROM AUTHOR]

We prove that the Weyl fractional derivative is a useful instrument to express certain properties of the zeta related functions. Specifically, we show that a known reflection property of the Hurwitz zeta function ζ(n, a) of integer first argument can be extended to the more general case of ζ(s, a), with complex s, by replacement of the ordinary derivative of integer order by Weyl fractional derivative of complex order. Besides, ζ(s, a) with ℜ(s) > 2 is essentially the Weyl (s − 2)-derivative of ζ(2, a). These properties of the Hurwitz zeta function can be immediately transferred to a family of polygamma functions of complex order defined in a natural way. Finally, we discuss the generalization of a recently unveiled reflection property of the Lerch's transcendent. [ABSTRACT FROM AUTHOR]

We prove a monotonicity property of the Hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. We provide a probabilistic interpretation of our result by exploiting a connection between Hurwitz zeta function and the cumulants of the beta-exponential distribution. [ABSTRACT FROM AUTHOR]

Recently, authors of Wu et al. (Adv. Comput. Math. 38:187-205, 2013) studied a Bernstein polynomial approximation scheme based on stochastic sampling and obtained a sixth-order moment estimate for the underlying random variable in terms of the modulus of continuity. In the current paper, we employ a new technique and establish estimates for all the even-order moments. Our work gives a strong indication that the probabilistic convergence rate of the stochastic Bernstein approximation is exponential with respect to the modulus of continuity, which we leave as a conjecture at the end of the paper. [ABSTRACT FROM AUTHOR]

We introduce the generalized zeta matrix function, digamma matrix function and polygamma matrix function. We obtain the regions of convergence of these matrix functions, some recurrence relations and their integral representations. An identity involving hypergeometric matrix function 3F2 in the form of digamma matrix function is also given. [ABSTRACT FROM AUTHOR]

The connection is considered between integrals and series involving polygamma ψ (z) and zeta ζ (z, s) functions, Bernoulli Bn (z) and Euler En (z) polynomials, and Bernoulli Bn and Euler En numbers. [ABSTRACT FROM AUTHOR]

In this paper, we find a solution of an open problem posed by Alzer, Berg, and Koumandos: determine ( α , m ) ∈ R + × N such that the function x α | ψ ( m ) ( x ) | is completely monotonic on ( 0 , ∞ ) , where ψ ( x ) denotes the logarithmic derivative of Euler’s gamma function.

In this paper, we introduce the notion of completely mixed monotonicity of a function of several variables, very few of which have appeared. We give a necessary and sufficient condition for a function constructed by ratios of gamma functions to be completely mixed monotonic. From this, some new inequalities for gamma, psi, and polygamma functions are derived. [ABSTRACT FROM AUTHOR]

In this paper, we present a general comparison theorem for two divided differences of a three times differentiable function. This gives a unified treatment for (logarithmically) complete monotonicity, monotonicity and inequalities involving some special functions including gamma, psi and polygamma functions. As their consequences, we not only refine and generalize some important results, but also present simple and interesting alternative proofs of certain earlier results. [ABSTRACT FROM AUTHOR]

The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, is generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application to sums of double zeta values. A set of identities for the Ramanujan and generalized Ramanujan polynomials is presented. An alternative proof of Lettington's identity is provided, together with its generalizations to the Hurwitz and Lerch zeta functions, hence to Dirichlet L series, to Eisenstein series, and to general Mellin transforms. The Hurwitz numbers H ˜ n occur in the Laurent expansion about the origin of a certain Weierstrass ℘ function for a square lattice, and are highly analogous to the Bernoulli numbers. An integral representation of the Laurent coefficients about the origin for general ℘ functions, and for these numbers in particular, is presented. As a Corollary, the asymptotic form of the Hurwitz numbers is determined. In addition, a series representation of the Hurwitz numbers is given, as well as a new recurrence. Other results concern the Matter numbers of the equianharmonic case of the ℘ function. [ABSTRACT FROM AUTHOR]

For high‐resolution synthetic aperture radar (SAR), constant false alarm rate (CFAR) detectors are widely used to separate targets from background and it is proved that generalised gamma distribution (GΓD) can model the non‐homogeneous clutter appropriately. However, CFAR detectors based on GΓD possess two problems. First, the methods for solving highly non‐linear equations of GΓD resort to numerical iterative algorithms, which are computationally expensive. To avoid this, the authors present a novel analytic solution for parameter estimation of GΓD by exploiting a third‐order approximation of the polygamma function. This novel analytic solution can result in more accurate parameter estimation and can fit a wider range of the parameters. The second problem is that in a multi‐target SAR image, some target pixels may be classified as clutter pixels. To select pixels of interest, an iterative sliding window approach is often used in CFAR. They analyse the relationship between the detection probability and the number of iterations, and prove that this strategy can reduce the miss rate and false detection rate effectively. On the basis of the aforementioned parameter estimation method and the iterative sliding window approach, an innovative and effective CFAR detector is proposed in this study and its superiority is demonstrated by experiments. [ABSTRACT FROM AUTHOR]

In this paper, we find rational zeta series within terms ofand, the Dirichlet beta function. We then develop a certain family of generalized rational zeta series using the generalized Clausen functionand use those results to discover a second family of generalized rational zeta series. As a special case of our results from Theorem 3.1, we prove a conjecture given in 2012 by F.M.S. Lima. Later on, we use the polygamma functionand negapolygammasto achieve the same three types of generalized rational zeta series but withon the numerator instead of. [ABSTRACT FROM PUBLISHER]

In the paper, the authors find a new explicit formula, the monotonicity, and the convexity of the average numbers of comparisons used by the quicksort. Moreover, the authors also present the absolute monotonicity of the generating function of the average numbers of comparisons used by the quicksort. [ABSTRACT FROM AUTHOR]

We give a meaning to the expression ψ ( n ) ( − m ) in neutrix setting. Further the incomplete gamma type function γ ⁎ ( α , x − ) is introduced for negative values of α . [ABSTRACT FROM AUTHOR]

In this paper, we establish some inequalities bounding the ratio Γp,k(x)/Γp,k(y), where Γp,k(.) is the (p,k)-analogue of the Gamma function. Consequently, some previous results are recovered from the obtained results. [ABSTRACT FROM AUTHOR]

We present improvements of approximation formula for Wallis ratio related to a class of inequalities stated in [D.-J. Zhao, On a two-sided inequality involving Wallis's formula, Math. Practice Theory, 34 (2004), 166-168], [Y. Zhao and Q. Wu, Wallis inequality with a parameter, J. Inequal. Pure Appl. Math., 7(2) (2006), Art. 56] and [C. Mortici, Completely monotone functions and the Wallis ratio, Applied Mathematics Letters, 25 (2012), 717-722]. Some sharp inequalities are obtained as a result of monotonicity of some functions involving gamma function. [ABSTRACT FROM AUTHOR]

In the paper, we establish necessary and sufficient conditions for two families of functions involving divided differences of the di- and tri-gamma functions to be completely monotonic. Consequently, we derive necessary and sufficient conditions for two families of functions involving the ratio of two gamma functions to be logarithmically completely monotonic. Furthermore, we deduce some inequalities for bounding the ratio of two gamma functions and divided differences of polygamma functions. [ABSTRACT FROM AUTHOR]

The Stieltjes constants $$\gamma _k(a)$$ appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function $$\zeta (s,a)$$ about $$s=1$$ . We present the evaluation of $$\gamma _1(a)$$ and $$\gamma _2(a)$$ at rational arguments, this being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for $$\gamma _0(a)$$ , $$\gamma _1(a)$$ , and $$\gamma _2(a)$$ , and point out that these formulas are cases of an addition formula previously presented. We present certain integral evaluations generalizing Gauss' formula for the digamma function at rational argument. In addition, we give the asymptotic form of $$\gamma _k(a)$$ as $$a \rightarrow 0$$ as well as a novel technique for evaluating integrals with integrands with $$\ln (-\ln x)$$ and rational factors. [ABSTRACT FROM AUTHOR]

We present asymptotic expansion of function involving the ratio of gamma functions and provide a recurrence relation for determining the coefficients of the asymptotic expansion. As a consequence, we obtain asymptotic expansion of the Wallis sequence. Also, we establish sharp inequalities for the Wallis sequence. [ABSTRACT FROM AUTHOR]

The paper is about calculating the following class of quadratic series:wherek≥2 is an integer. We show thatSkcan be expressed as a rational linear combination of zeta function values, binomial coefficients and products of two zeta function values. [ABSTRACT FROM AUTHOR]