Evaluations of nonlinear Euler sums of weight ten

In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral representations of series. Then we apply it to obtain the closed forms of all quadratic Euler sums of weight equal to ten. Furthermore, we also establish some relations between multiple zeta (star) values and nonlinear Euler sums. As applications of these relations, we give new closed form representations of several cubic Euler sums through single zeta values and linear sums. Finally, with the help of numerical computations of Mathematica or Maple, we evaluate several other Euler sums of weight ten

Evaluations of Euler type sums of weight ≤\leq 5

Let $p,p_1,\ldots,p_m$ be positive integers with $p_1\leq p_2\leq\cdots\leq p_m$ and $x\in [-1,1)$, define the so-called Euler type sums ${S_{{p_1}{p_2} \cdots {p_m},p}}\left( x \right)$, which are the infinite sums whose general term is a product of harmonic numbers of index $n$, a power of $n^{-1}$ and variable $x^n$, by $$S_{p_1 p_2 \cdots p_m, p}(x) := \sum_{n = 1}^\infty \frac{H_n^{(p_1)} H_n^{(p_2)} \cdots H_n^{(p_m)}} {n^p} x^n \quad (m\in \mathbb{N} := \{1,2,3,\ldots\}),$$ where $H_n^{(p)}$ is defined by the generalized harmonic number. Extending earlier work about classical Euler sums, we prove that whenever $p+p_1+\cdots+p_m \leq 5$, then all sums ${S_{{p_1}{p_2} \cdots {p_m},p}}\left( 1/2\right)$ can be expressed as a rational linear combination of products of zeta values, polylogarithms and $\log(2)$. The proof involves finding and solving linear equations which relate the different types of sums to each other

Euler Sums of Hyperharmonic Numbers

The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: {\sigma}(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of series of Hurwitz zeta function values. This is a generalization of a result of Mez\H{o} and Dil. We also provide an explicit evaluation of {\sigma}(r,m) in a closed form in terms of zeta values and Stirling numbers of the first kind. Furthermore, we evaluate several other series involving hyperharmonic numbers.Comment: 9 page

On one dimensional digamma and polygamma series related to the evaluation of Feynman diagrams

We consider summations over digamma and polygamma functions, often with summands of the form (\pm 1)^n\psi(n+p/q)/n^r and (\pm 1)^n\psi^{(m)} (n+p/q)/n^r, 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.Comment: to appear in J. Comput. Appl. Math.; corrected proof available online with this journal; no figure

On generalized harmonic numbers, Tornheim double series and linear Euler sums

Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a linear combination of Tornheim double series of the same weight. New closed form evaluations of various Euler sums are presented. Finally certain combinations of linear Euler sums that are reducible to Riemann zeta values are discovered.Comment: Corrected typos, added theorem

Relating log-tangent integrals with the Riemann zeta function

We show that integrals involving log-tangent function, with respect to certain square-integrable functions on $(0, \pi/2)$, can be evaluated by some series involving the harmonic number. Then we use this result to establish many closed forms relating to the Riemann zeta function at odd positive integers. In addition, we show that the log-tangent integral with respect to the Hurwitz zeta function defines a meromorphic function and that its values depend on the Dirichlet series $\zeta_h(s) :=\sum_{n = 1}^\infty h_n n^{-s}$, where $h_n = \sum_{k=1}^n(2k-1)^{-1}$.Comment: 20 page

Resolution of Some Open Problems Concerning Multiple Zeta Evaluations of Arbitrary Depth

We prove some new evaluations for multiple polylogarithms of arbitrary depth. The simplest of our results is a multiple zeta evaluation one order of complexity beyond the well-known Broadhurst-Zagier formula. Other results we provide settle three of the remaining outstanding conjectures of Borwein, Bradley, and Broadhurst. A complete treatment of a certain arbitrary depth class of periodic alternating unit Euler sums is also given.Comment: 21 pages, To appear in Compositio Mathematic

Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter

We continue the study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we show the following results: Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth (see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions. Theorem B: The epsilon expansion of a hypergeometric function with one half-integer value of parameter (see Eq. (1.2)) is expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are ratios of polynomials. Some extra materials are available via the www at this http://theor.jinr.ru/~kalmykov/hypergeom/hyper.htmlComment: 24 pages, latex with amsmath and JHEP3.cls; v2: some typos corrected and a few references added; v3: few references added

A Comment on Matiyasevich's Identity #0102 with Bernoulli Numbers

We connect and generalize Matiyasevich's identity #0102 with Bernoulli numbers and an identity of Candelpergher, Coppo and Delabaere on Ramanujan summation of the divergent series of the infinite sum of the harmonic numbers. The formulae are analytic continuation of Euler sums and lead to new recursion relations for derivatives of Bernoulli numbers. The techniques used are contour integration, generating functions and divergent series.Comment: 10 page

Conjectured Enumeration of irreducible Multiple Zeta Values, from Knots and Feynman Diagrams

Multiple zeta values (MZVs) are under intense investigation in three arenas -- knot theory, number theory, and quantum field theory -- which unite in Kreimer's proposal that field theory assigns MZVs to positive knots, via Feynman diagrams whose momentum flow is encoded by link diagrams. Two challenging problems are posed by this nexus of knot/number/field theory: enumeration of positive knots, and enumeration of irreducible MZVs. Both were recently tackled by Broadhurst and Kreimer (BK). Here we report large-scale analytical and numerical computations that test, with considerable severity, the BK conjecture that the number, $D_{n,k}$, of irreducible MZVs of weight $n$ and depth $k$, is generated by $\prod_{n\ge3}\prod_{k\ge1}(1-x^n y^k) ^{D_{n,k}}=1-\frac{x^3y}{1-x^2}+\frac{x^{12}y^2(1-y^2)}{(1-x^4)(1-x^6)}$, which is here shown to be consistent with all shuffle identities for the corresponding iterated integrals, up to weights $n=44, 37, 42, 27$, at depths $k=2, 3, 4, 5$, respectively, entailing computation at the petashuffle level. We recount the field-theoretic discoveries of MZVs, in counterterms, and of Euler sums, from more general Feynman diagrams, that led to this success.Comment: 11 pages, LaTe