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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]

We introduce a generic inequality of the form f 2 (x) ≤ f (s) f (2 x − s) and show that many special functions such as gamma and polygamma functions, Riemann zeta function, beta and incomplete beta functions, Gauss and confluent hypergeometric functions, elliptic integrals and error function satisfy such type of the inequality. We also obtain new inequalities of this type for inverse trigonometric functions arcsin ⁡ x and arctan ⁡ x. [ABSTRACT FROM AUTHOR]

We establish some identities of Euler related sums. By using these identities, we discuss the closed form representations of sums of harmonic numbers and reciprocal parametric binomial coefficients through parametric harmonic numbers, shifted harmonic numbers and Riemann zeta function with positive integer arguments. In particular, we investigate products of quadratic and cubic harmonic numbers and reciprocal parametric binomial coefficients. Some illustrative special cases as well as immediate consequences of the main results are also considered. [ABSTRACT FROM AUTHOR]

In this paper we establish the modular relation for Maass forms to the effect that the Fourier–Whittaker expansion and the ramified functional equation are equivalent, i.e. the RHB (Riemann–Hecke–Bochner) correspondence. This arises from the new standpoint that Maass' procedure is one of the ways leading to the Fourier–Whittaker expansion similar to the situation that various representations of the modified Bessel function K ν have provided such vast amount of different methods including the Hardy transform and the beta-transform leading to the Fourier-Bessel expansion from opposite directions. Naturally, we may abridge the gap between Hecke Eisenstein series and Epstein zeta-functions and the Chowla–Selberg integral formula and Ramanujan–Guinand formula are immediate corollaries, whose remark is due to the referee. [ABSTRACT FROM AUTHOR]

An investigation into families of integrals containing powers of the arctanh and log functions will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals. In another special case we prove that the corresponding log-tanh integral can be represented as a linear combination of the product of zeta functions and the Dirichlet eta function. [ABSTRACT FROM AUTHOR]

In this paper, we give explicit evaluation for some integrals involving polylogarithm functions of types ∫ 0 x t m L i p (t) d t and ∫ 0 x log m ⁡ (t) L i p (t) d t. Some more integrals involving the logarithm function will also be derived. [ABSTRACT FROM AUTHOR]

The (generalized) Mellin transforms of Gegenbauer polynomials have polynomial factors p n λ (s) , whose zeros all lie on the 'critical line' ℜ s = 1 / 2 (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2, and S:3/1 binomial coefficient forms. Their 'critical polynomial' factors are then identified in terms of 3 F 2 (1) hypergeometric functions. Furthermore, we extend these results to a one-parameter family of critical polynomials that possess the functional equation p n (s ; β) = ± p n (1 − s ; β). Normalization yields the rational function q n λ (s) whose denominator has singularities on the negative real axis. Moreover, as s → ∞ along the positive real axis, q n λ (s) → 1 from below. For the Chebyshev polynomials, we obtain the simpler S:2/1 binomial form, and with C n the nth Catalan number, we deduce that 4 C n − 1 p 2 n (s) and C n p 2 n + 1 (s) yield odd integers. The results touch on analytic number theory, special function theory, and combinatorics. [ABSTRACT FROM AUTHOR]

Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series. [ABSTRACT FROM AUTHOR]

In this paper we develop new identities for Euler sums with rational argument. We shall investigate and report on new Euler identities of weight for m,p,q positive integers and m + p + q an even integer, but with a non-unitary argument of the harmonic numbers. Some examples of these Euler identities will be given in terms of Riemann zeta values, Dirichlet values and other special functions. We shall also give an explicit solution to an integral with a log polylog integrand. [ABSTRACT FROM AUTHOR]

The Cauchy–Schlömilch transformation states that for a function f and , the integral of and over the interval are the same. This elementary result is used to evaluate many non-elementary definite integrals, most of which cannot be obtained by symbolic packages. Applications to probability distributions are also given. [ABSTRACT FROM AUTHOR]

In Ferreira and López [Asymptotic expansions of the Hurwitz–Lerch zeta function. J Math Anal Appl. 2004;298(1):210–224], the authors derived an asymptotic expansion of the Lerch's transcendent for large , valid for , and. In this paper, we study the special case not covered in Ferreira and López [Asymptotic expansions of the Hurwitz–Lerch zeta function. J Math Anal Appl. 2004;298(1):210–224], deriving a complete asymptotic expansion of the Lerch's transcendent for z>1 and as goes to infinity. We also show that when a is a positive integer, this expansion is convergent for. As a corollary, we get a full asymptotic expansion for the sum for fixed as. Some numerical results show the accuracy of the approximation. [ABSTRACT FROM AUTHOR]

In this paper, by using residue method, we obtain the representations of some basic linear generalized Euler sums with parameters. Based on the linear generalized Euler sums with parameters, some new Euler sums are obtained and expressed in the closed forms. When the parameters of new Euler sums take special values, we can get some usual expressions of Euler sums. Moreover, the integrals of many special functions can be expressed as the Euler sums given in this paper. [ABSTRACT FROM AUTHOR]

Let p,x be real numbers, and s be a complex number, with R(s) > 1 - r, p ≥ 1, and x+1>0. The zeta function Zαp (s; x) is defined by ..., where α = (α1, ..., αr) is a r-tuple positive integers, and Liα(z) is the one-variable multiple polylogarithms. Since Zα1 (s; 0) = ξ(α; s), we call this function as a generalized Arakawa-Kaneko zeta function. In this paper, we investigate the properties and values of Zαp (s; x) with different values s, x, and p. We then give some applications on them. [ABSTRACT FROM AUTHOR]

In this paper, by choosing different kernel functions and base functions, we obtain some Euler sums with parameters. Moreover, we also obtain the new Euler sums with parameters by differentiating, limiting and elementary arithmetic. Thus, more Euler sums with parameters can be obtained. Furthermore, some Euler sums given in this paper are closed forms. [ABSTRACT FROM AUTHOR]

Using known theta identities and formulas of S. Ramanujan and G. Hardy among others we prove several formulas for the Riemann zeta-function and two Dirichlet series. [ABSTRACT FROM AUTHOR]

We prove several relations on multiple Hurwitz-Riemann zeta functions. Using analytic continuation of these multiple Hurwitz-Riemann zeta functions, we quote at negative integers Euler's nonlinear relation for generalized Bernoulli polynomials and numbers. As an application, we give a general convolution identity for Bernoulli numbers. [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]

Jacobian elliptic functions are used to obtain formulas for deriving indefinite integrals for the Jacobi Zeta function and Heuman's Lambda function. Only sample results are presented, mostly obtained from powers of the twelve Glaisher elliptic functions. However, this sample includes all such integrals in the literature, together with many new integrals. The method used is based on the differential equations obeyed by these functions when the independent variable is the argumentuof elliptic function theory. The same method was used recently, in a companion paper, to derive similar integrals for the three canonical incomplete elliptic integrals. [ABSTRACT FROM AUTHOR]

By reformulating four hypergeometric series formulae, we derive 36 Apéry-like series expressions for the Riemann zeta function, including a couple of identities conjectured by Sun [New series for some special values of L-functions. Nanjing Univ J Math. 2015;32(2): 189–218]. [ABSTRACT FROM AUTHOR]

In this note, we show that the values of integrals of the log-tangent function with respect to any square-integrable function onmay be determined (or approximated) by an infinite (or finite) sum involving the Riemann zeta function at odd positive integers. [ABSTRACT FROM AUTHOR]

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an integral representation for the finite parts at the poles. Similar results are derived for an associated function, which we term homogeneous Barnes zeta function. Our expressions immediately yield analogous representations for the logarithm of the Barnes gamma function, including the particular case also known as multiple gamma function. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some formulae related to Euler Sums and give the closed-form representations for the sumwhich can be evaluated in terms of the Riemann zeta functions and generalized harmonic numbers. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the Turán-type inequalities for Struve functions, modified Struve functions, Anger–Weber functions and Hurwitz zeta function, by using a new form of the Cauchy–Bunyakovsky–Schwarz inequality. [ABSTRACT FROM AUTHOR]

In this paper, by using the method of partial fraction decomposition and integral representations of series, we establish some expressions of series involving harmonic numbers and binomial coefficients in terms of zeta values and harmonic numbers. Furthermore, we can obtain some closed form representations of sums of products of quadratic (or cubic) harmonic numbers and reciprocal binomial coefficients, and some explicit evaluations are given as applications. The given representations are new. [ABSTRACT FROM AUTHOR]

The asymptotic behaviour of the Chebyshev–Stirling numbers of the second kind, a special case of the Jacobi–Stirling numbers, has been established in a recent paper by Gawronski, Littlejohn and Neuschel. In this paper, we provide an asymptotic formula for the Chebyshev–Stirling numbers of the first kind. New recurrence relations for the Euler–Riemann zeta functionare derived in this context. [ABSTRACT FROM PUBLISHER]

The paper is about calculating in closed form the following multiple fractional part integralwhere,are integers anddenotes the fractional part ofx. We prove that the integralcan be expresses in terms of a difference of two rational numbers and a series involving Riemann zeta function values and some binomial coefficients. [ABSTRACT FROM AUTHOR]

In this paper, the following multiple fractional part integralsandare calculated for non-negative integersand positive integern, wheredenotes the fractional part ofuandIt is proved thathas a closed form for non-negative integer. We also prove that(and) andcan be expressed as linear combinations of the Riemann zeta function, logarithmic function and some binomial coefficients. In particular,has a closed form for. Moreover, some identities and recursive formulas of the above integrals are obtained. [ABSTRACT FROM PUBLISHER]

By using a linear operator associated with theλ-generalized Hurwitz–Lerch zeta function, which is defined here by means of the Hadamard product (or convolution), the authors introduce and investigate various properties of certain subclasses of meromorphically univalent functions in the punctured unit disk. Some closely related (known or new) corollaries and consequences of the main results presented in this paper are also considered. [ABSTRACT FROM AUTHOR]

This paper discusses generalization of some logarithmic integrals appeared in Adamchik, Medina and Moll's works. [ABSTRACT FROM AUTHOR]

In this short note, we point out an interesting connection between series with zeta values and certain infinite products. Using this connection we give a closed-form evaluation of a special class of series with zeta values in the coefficients. [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]

We develop families of generalized hypergeometric function identities at argument±1 in terms of generalized harmonic numbers and Hurwitz zeta functions. [ABSTRACT FROM AUTHOR]

A generalized theorem related to harmonic sums is investigated from which we are able to represent an infinite number of sums of harmonic numbers in both multiple integral and closed forms. The new theorem generalizes known existing results in the published literature. [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 this paper we consider the following Euler sumswherepandqare positive integers,zandzi(i=1, 2, … ) are complex numbers on the unit circle or points within the unit circle except the origin andxis a complex number with Re x>0. We introduce a kernel function with a complex parameter and use it to obtain some identities of the linear extended Euler sums with parameters and some new Euler sums, which generalize the identities of the standard Euler sums. The linear extended Euler sums discussed in this paper can be expressed in terms of the Lerch transcendent. In some cases they can also be expressed by Riemann zeta function and Hurwitz zeta function. [ABSTRACT FROM PUBLISHER]

In this paper, some identities on the Hurwitz zeta function and the following extended Euler sumsare given, where. Furthermore, ζ (2n+1, p/q) can be expressed as the Riemann zeta functions whenp/q=⅓, ⅔, ¼, ¾, ⅙, ⅚. [ABSTRACT FROM AUTHOR]

The paper is about calculating a special class of fractional part integrals of the formwherenis a positive integer andxdenotes the fractional part ofx. We show that these integrals can be expressed as rational linear combination of Riemann zeta function values and a series involving the difference of two zeta function values of consecutive integers. [ABSTRACT FROM AUTHOR]

We investigate sums of reciprocal binomial coefficients with quadratic harmonic numbers and harmonic numbers of order two. We create new analytical identities in closed form of Euler-related sums. [ABSTRACT FROM PUBLISHER]

By using the theory of the Riemann zeta-function, Mellin's and index-convolution transforms we prove a summation formula involving sums of the form ∑ σiτ(n)n−iτ/2f(n), where σμ(n) is the sum of the μth powers of the divisors ofn. Here iτ, τ∈ℝ, is a pure imaginary complex number. When tau=0 it drives to the well- known Voronoi's formula. Some examples of summation formulas such as Voronoi's, Koshliakov's, Ramanujan's, Guinand's, Nasim's are demonstrated. Finally, a relationship between the Kontorovich–Lebedev and Tichmarsh transforms is obtained. [ABSTRACT FROM PUBLISHER]

In this paper, we show an analog of hybrid universality theorem (see ?. Pa?kowski,Hybrid joint universality theorem for Dirichlet L-functions, Acta Arith. 141(1) (2010), pp. 59–72) for L-functions without Euler product like Lerch zeta-functions and periodic Hurwitz zeta-functions. More precisely, we prove that any analytic functions can be approximated by some L-functions shifted by i? and, simultaneously, finitely many real numbers can be approximated by ?1?, …,?n?, where ?1, …,?nare real numbers linearly independent over ?. [ABSTRACT FROM PUBLISHER]

The main purpose of this paper is to construct generating functions for some families of Genocchi type polynomials. By applying the derivative operator to these generating functions, we define a function which interpolates these polynomials at negative integers. Among other results, we prove a multiplication theorem for these polynomials. Furthermore, we give several other identities and applications associated with these and related polynomials and their interpolation functions. [ABSTRACT FROM PUBLISHER]

In the paper, a multiplicative version of the distribution of the vector (|ζ(s)|, ζ(s)), where ζ(s) is the Riemann zeta-function, is considered and probabilistic limit theorems are obtained. The dependence between |ζ(s)| and ζ(s) in terms of m-characteristic transforms is also discussed. [ABSTRACT FROM AUTHOR]

Series arising from Volterra integral equations of the second kind are summed. The series involve inverse powers of roots of the characteristic equation. It is shown how previous similar series obtained from differential–difference equations are particular cases of the present development. A number of novel and interesting results are obtained. The techniques are demonstrated through illustrative examples. The application of general techniques to obtain expressions involving the zeros of the Riemann zeta function is investigated. In particular, novel two-parameter expressions are obtained involving the non-trivial zeros of the zeta function. [ABSTRACT FROM AUTHOR]

In a recent work, Dancs and He found an Euler-type formula for ζ(2n+1), n being a positive integer, which contains a series they could not reduce to a finite closed form. This open problem reveals a greater complexity in comparison with ζ(2n), which is a rational multiple of π2n . For the Dirichlet beta function, the things are ‘inverse’: β(2n+1) is a rational multiple of π2n+1 and no closed-form expression is known for β(2n). Here in this work, I modify the Dancs–He approach in order to derive an Euler-type formula for β(2n), including β(2)=G, Catalan's constant. I also convert the resulting series into zeta series, which yields new exact closed-form expressions for a class of zeta series involving β(2n) and a finite number of odd zeta values. A closed-form expression for a certain zeta series is also conjectured. [ABSTRACT FROM AUTHOR]

In this paper, we shall locate Plana's summation formula à la Koshlyakov as a form of the modular relation for the Riemann zeta-function and an analytic continuation technique furnished by Lemma 1. Then by Plana's summation formula, we shall prove an important integral representation for the Hurwitz–Lerch zeta-function through confluent hypergeometric functions, which cover a wide range of integral representations attributed to famous mathematicians. We shall also locate the genesis of Mikolás’ formula and prove that the functional equations for the Hurwitz zeta-function and the Riemann zeta-function are equivalent. [ABSTRACT FROM PUBLISHER]

In this paper, making use of the generalized Srivastava–Attiya operator , we introduce a new subclass of spiral-like functions and discuss some subordination results for functions in this generalized function class. Furthermore, we mention some known and new results, which follow as special cases of our results. [ABSTRACT FROM PUBLISHER]

From the Kummer 2 F 1-summation theorem, the Dixon–Kummer 4 F 3-summation theorem and the Dougall–Dixon 5 F 4-summation theorem, we establish, by means of the Bell polynomials, three general formulas related to the generalized harmonic numbers and the Riemann zeta function. Based on these three general formulas, we further find series of harmonic number identities. Some of these identities involve both finite summation and infinite series, so that we can determinate the explicit expressions of numerous infinite series. In particular, we show that several interesting analogues of the Euler sums can be evaluated. [ABSTRACT FROM AUTHOR]

By using the theory of the Mellin and Mellin convolution type transforms, we prove a general summation formula of Voronoi involving sums of the form ∑ d k (n)f(n), where d k (n), k=2, 3, …, d 2(n)≡ d(n) is the number of ways of expressing n as a product of k factors. These sums are related to the famous Dirichlet divisor problem of determining the asymptotic behaviour as x→∞ of the sum D k (x)=∑ n≤x d k (n). In particular, we generalize Koshliakov's formula and certain identities from Ramanujan's lost notebook to the case of hyper-Bessel functions and Jacobi elliptic theta functions. New examples of Voronoi's summation formulas involving Bessel, exponential functions and their products, which are based on a comprehensive Marichev's table of Mellin's transforms are given. The equivalence of these relations to the functional equation for the Riemann Zeta -function is discussed. An extension of the Koshliakov formula involving the Kontorovich–Lebedev transform is obtained. [ABSTRACT FROM AUTHOR]

Motivated essentially by their potential for applications in a wide range of mathematical and physical problems, the log-sine integrals have been evaluated, in the existing literature on the subject, in many different ways. The main object of this paper is to present explicit evaluations of some families of log-sine integrals by making use of a familiar algorithm, and also those of log-cosine and other related integrals by using some known integral formulas expressed in terms of the Gamma functions. [ABSTRACT FROM AUTHOR]

We obtain the asymptotics for the fourth power moment of the periodic zeta-function with rational parameter λ. [ABSTRACT FROM AUTHOR]