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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]

In this paper, the partial derivatives Bp,q(x, y) = ∂q + p/∂xp∂yq B(x, y) of the Beta function B(x, y) are expressed in terms of a finite number of the Polygamma function, where p and q are non-negative integers, x and y are complex numbers. In particular, Bp,q(x; y) can be expressed by the Riemann zeta function if x is equal to n or n + 1/2 and y is equal to m or m + 1/2, where n and m are integers. Furthermore, many integral functions associated with B(x, y) and Bp,q(x, y) can be expressed as the closed forms. [ABSTRACT FROM AUTHOR]

The multiple gamma function Γn(z), defined by a recurrence-functional equation as a generalization of the Euler gamma function, is used in many applications of pure and applied mathematics, and theoretical physics. The theory of the multiple gamma function has been related to certain spectral functions in mathematical physics, to the study of functional determinants of Laplacians of then-sphere, to HeckeL-functions, to the Selberg zeta function, and to the random matrix theory. There is a wide class of definite integrals and infinite sums appearing in statistical physics (the Potts model) and the lattice theory, which can be computed by means of the Γn(z) function. This paper presents new integral representations for the multiple gamma function and other mathematical functions and constants. [ABSTRACT FROM AUTHOR]

In the paper the authors show that neither the function ... nor its negative is completely monotonic on (0,∞), where ai >0 and bi ≥0 are real numbers, i ≥2, Γ(x) is the classical Euler's gamma function, and ... are polygamma functions. Moreover, some other results are given in the form of remarks. [ABSTRACT FROM AUTHOR]

In the paper, the authors construct three new sequences converging fast to Euler-Mascheroni constant γ and obtain a sharp double inequality for bounding the harmonic number. [ABSTRACT FROM AUTHOR]

Necessary 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. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we give a sufficient condition and a necessary and sufficient condition for a function class associated with the gamma function to be logarithmically completely monotonic. Some applications are also presented. [Copyright &y& Elsevier]

In the paper, some lower bounds for polygamma functions are refined. Moreover, several open problems are posed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, an efficient statistical model, called generalized Gamma distribution (G \Gamma D), for the empirical modeling of synthetic aperture radar (SAR) images is proposed. The G \Gamma D 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 G \Gamma D 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 G \Gamma D convenient for applications in the online SAR image processing. Finally, experimental results from tests carried out with actual SAR images demonstrate that the G \Gamma D can achieve better goodness of fit than the state-of-the-art pdfs. [ABSTRACT FROM PUBLISHER]

Let log (d/dx) be the generator of the1-parameter group {da dxaȣa ∈ ℝ} of fractional order differentiations acting on the space of operators of Mikusinski ([5]). The Lie algebra glog generated by log (d/dx) and log x is a deformation and can be regarded as the logarithm of Heisenberg Lie algebra. We show glog is isomorphic to the Lie algebra generated by d / ds log(Γ(1 + s)) and d / ds. Hence as a module, 햼log is isomorphic to the module generated by d/ds and polygamma functions. Structure of the group generated by 1-parameter groups {da/dxa ȣa ∈ ℝ} and {xaȣa ∈ ℝ}, is also determined. [ABSTRACT FROM AUTHOR]

Abstract: We evaluate generalized harmonic number sums with parameter in terms of values of polylogarithm functions, and several examples are given. Instances of such sums occur in diverse areas including analytic number theory, and in calculations of high energy, nuclear, and atomic physics. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, we establish necessary and sufficient conditions for the functions and respectively to be monotonic and completely monotonic on , where , and are scalars, and are polygamma functions. [Copyright &y& Elsevier]

We present some logarithmically complete monotonicity and logarithmically absolute monotonicity properties for the gamma function. These extend some known results. [ABSTRACT FROM AUTHOR]

Abstract: A simple class of algorithms for the efficient computation of the Hurwitz zeta and related special functions is given. The algorithms also provide a means of computing fundamental mathematical constants to arbitrary precision. A number of extensions as well as numerical examples are briefly described. The algorithms are easy to implement and compete with Euler–Maclaurin summation-based methods. [Copyright &y& Elsevier]

Let q be a natural number and $f:\mathbb{Z}/q\mathbb{Z} \rightarrow \overline{\mathbb{Q}}$. We consider the Dirichlet series ∑n ≥ 1 f(n)/ns and relate its value when s is a natural number, to the special values of the polygamma function. For certain types of functions f, we evaluate the special value explicitly and use this to study linear independence over ℚ of L(k,χ) as χ ranges over Dirichlet characters mod q which have the same parity as k. [ABSTRACT FROM AUTHOR]

Abstract: In the article, sufficient and necessary conditions that a class of functions involving ratio of Euler’s gamma functions and originating from Wendel’s and Gautschi’s inequalities are logarithmically completely monotonic are presented. From this, Wendel’s, Gautschi’s, Kershaw’s, Laforgia’s, Bustoz-Ismail’s, Merkle’s and Elezović-Giordano-Pečarić’s inequalities are refined, extended and sharpened, and a double inequality on the divided differences of the psi and polygamma functions is deduced straightforwardly. [Copyright &y& Elsevier]

Abstract: In the paper, a new upper bound in the second Kershaw''s double inequality involving ratio of gamma functions is established, and, as generalizations of the second Kershaw''s double inequality, the divided differences of the psi and polygamma functions are bounded. [Copyright &y& Elsevier]

Abstract: A variety of infinite series representations for the Hurwitz zeta function are obtained. Particular cases recover known results, while others are new. Specialization of the series representations apply to the Riemann zeta function, leading to additional results. The method is briefly extended to the Lerch zeta function. Most of the series representations exhibit fast convergence, making them attractive for the computation of special functions and fundamental constants. [Copyright &y& Elsevier]

Abstract: In this article, the authors present a necessary condition, a sufficient condition and a necessary and sufficient condition for a class of functions associated with the gamma function to be logarithmically completely monotonic. As a consequence of these results, supplements to the recent investigation by Chen and Qi [C.-P. Chen, F. Qi, Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl. 321 (2006) 405–411] are provided and a new Kečkić–Vasić type inequality is concluded. [Copyright &y& Elsevier]

For given real numbers a≥0, b∈ and c∈, let Fa, b, c(x)=[Γ(x+1)]1/x(1+a/x)x+b/xc and φa, b, c(x)=ψ''(x)+[2+(b+c)x-2x2]/x3+[3a(2a-b)+(6a-b)x+2x2]/(x+a)3 with x∈(0, ∞), where Γ(x) and ψ(x) are the well-known Euler gamma function and the psi or digamma function, respectively. In this article, it is revealed that the function Fa, b, c(x) for 2a≤3b≤-3c and its reciprocal 1/Fa, b, c(x) for 2a≤3b and 1+2b+c≥0 are logarithmically completely monotonic in (0, ∞), while the function φa, b, c(x) for 0≤2a≤3b and 1+2b+c≥0 and its negative-φa, b, c(x) for 0≤2a≤3b and b+c≤0 are completely monotonic in (0, ∞). [ABSTRACT FROM AUTHOR]

Abstract: It is well known that the standard Laplace distribution arises by taking the difference of two independent exponential random variables. In this note, a class of generalized Laplace distributions is introduced by taking the difference of two independent exponentiated exponential variables. The various properties of this class of generalized Laplace distributions, which are presented here, include its stochastic representations and moments. [Copyright &y& Elsevier]

Abstract: The authors apply a classical series identity involving the psi (or digamma) function with a view to deriving series representations for a number of known mathematical constants. Several closely-related consequences and results are also considered. [Copyright &y& Elsevier]

Abstract: We consider summations over digamma and polygamma functions, often with summands of the form and , where m, p, q, and r are positive integers. We develop novel general integral representations and present explicit examples. Special cases of the sums reduce to known linear Euler sums. The sums of interest find application in quantum field theory, including evaluation of Feynman amplitudes. [Copyright &y& Elsevier]

Abstract: By using the first Binet''s formula the strictly completely monotonic properties of functions involving the psi and polygamma functions are obtained. As direct consequences, two inequalities are proved. As an application, the best lower and upper bounds of the nth harmonic number are established. [Copyright &y& Elsevier]

We study the properties of a function ψ( z , q ) (the generalized polygamma function), intimately connected with the Hurwitz zeta function and defined for complex values of the variables z and q , which is entire in the variable z and reduces to the usual polygamma function ψ ( m ) ( q ) for z a non-negative integer m , and to the balanced negapolygamma function ψ (- m ) ( q ) introduced in Ref. [5] for z a negative integer - m . [ABSTRACT FROM AUTHOR]

We evaluate a variety of log-cosine integrals, one of which has value a rational multiple of πζ(4), while others evaluate to the form π[aζ(6)+b(3/4)ζ2(3)], where a>0 is rational and b is 1 or 2, and ζ is the Riemann zeta function. We show that these results may be obtained starting from a known tabulated result. The derivation yields results for certain classes of logarithm-trigonometric integrals, and indicates how further analytic results on digamma series may be obtained. [Copyright &y& Elsevier]

Recently a new distribution, named as generalized exponential distribution or exponentiated exponential distribution was introduced and studied quite extensively by the authors. It is observed that the generalized exponential distribution can be used as an alternative to the gamma distribution in many situations. Different properties like monotonicity of the hazard functions and tail behaviors of the gamma distribution and the generalized exponential distribution are quite similar in nature, but the later one has a nice compact distribution function. It is observed that for a given gamma distribution there exists a generalized exponential distribution so that the two distribution functions are almost identical. Since the gamma distribution function does not have a compact form, efficiently generating gamma random numbers is known to be problematic. We observe that for all practical purposes it is possible to generate approximate gamma random numbers using generalized exponential distribution and the random samples thus obtained cannot be differentiated using any statistical tests. Moreover, if there is a skewed data set where gamma distribution fits very well, the generalized exponential distribution also can be used. We use two real life data sets and observe that the fitted distribution functions are “almost identical” in many respects in both the cases. [ABSTRACT FROM AUTHOR]