Alternating Euler Sums Research Articles

Variant Alternating Euler Sums of Higher Order

Variant Alternating Euler Sums of Higher Order

Multiple zeta star values on 3–2–1 indices

In 2008, Muneta found explicit evaluation of the multiple zeta star value \(\zeta ^\star (\{3, 1\}^d)\), and in 2013, Yamamoto proved a sum formula for multiple zeta star values on 3–2–1 indices. In this paper, we provide another way of deriving the formulas mentioned above. It is based on our previous work on generating functions for multiple zeta star values and also on constructions of generating functions for restricted sums of alternating Euler sums. As a result, the formulas obtained are simpler and computationally more effective than the known ones. Moreover, we give explicit evaluations of \(\zeta ^\star (\{\{2\}^m, 3, \{2\}^m, 1\bigr \}^d)\) and \(\zeta ^\star (\{\{2\}^m, 3, \{2\}^m, 1\}^d, \{2\}^{m+1})\), which are new and have not appeared in the literature before.

On a result of Koecher concerning Markov–Apéry-type formulas for the Riemann zeta function

Koecher in 1980 derived a method for obtaining identities for the Riemann zeta function at odd positive integers, including a classical result for [Formula: see text] due to Markov and rediscovered by Apéry. In this paper, we extend Koecher’s method to a very general setting and prove two more specific but still rather general results. As applications, we obtain infinite classes of identities for alternating Euler sums, further Markov–Apéry-type identities, and identities for even powers of [Formula: see text].

Alternating Euler sums and BBP-type series

Alternating Euler sums and BBP-type series

Euler Sums and Integral Connections

In this paper, we present some Euler-like sums involving partial sums of the harmonic and odd harmonic series. First, we give a brief historical account of Euler’s work on the subject followed by notations used in the body of the paper. After discussing some alternating Euler sums, we investigate the connection of integrals of inverse trigonometric and hyperbolic type functions to generate many new Euler sum identities. We also give some new identities for Catalan’s constant, Apery’s constant and a fast converging identity for the famous ζ ( 2 ) constant.

Log-sine-polylog integrals and alternating Euler sums

The log-sine-polylog integrals were introduced by J. M. Borwein and A. Straub during their studies on special values of the log-sine integrals. In this paper we evaluate some of the log-sine-polylog integrals and show that they are intimately related to alternating Euler sums. These evaluations result in some interesting relations with respect to the first integral moments of the square of the Clausen function Cl2.

Explicit formulas of Euler sums via multiple zeta values

Flajolet and Salvy pointed out that every Euler sum is a Q-linear combination of multiple zeta values. However, in the literature, there is no formula completely revealing this relation. In this paper, using permutations and compositions, we establish two explicit formulas for the Euler sums, and show that all the Euler sums are indeed expressible in terms of MZVs. Moreover, we apply this method to the alternating Euler sums, and show that all the alternating Euler sums are reducible to alternating MZVs. Some famous results, such as the Euler theorem, the Borwein–Borwein–Girgensohn theorems, and the Flajolet–Salvy theorems can be obtained directly from our theory. Some other special cases, such as the explicit expressions of Srm,q, Srm,q¯, Sr¯m,q and Sr¯m,q¯, are also presented here. The corresponding Maple programs are developed to help us compute all the sums of weight w≤11 for non-alternating case and of weight w≤6 for alternating case.

Generating functions for multiple zeta star values

We study generating functions for multiple zeta star values in general form. These generating functions provide a connection between multiple zeta star values and multiple Euler sums, which allows us to express each multiple zeta star value in terms of multiple alternating Euler sums, and specifically, reduce the length of blocks of twos in the resulting sums.

A reduction formula for length-two polylogarithms and some applications

We use shuffle and stuffle relations to give a simple proof of a reduction formula for length-two multiple polylogarithms evaluated in complex parameters of absolute value 1 in terms of a finite sum of products of lengthone polylogarithms. This result was originally due to Nakamura and recently reproved by Panzer by different methods. This generalises results of Borwein and Girgensohn for alternating Euler sums and for multiple zeta values twisted by fourth roots of unity by the first author. We also explore implications for other colored multiple zeta values and present some applications to Mahler measure and Feynman diagrams.

Restricted sum formula of alternating Euler sums

In this paper, we study restricted sum formulas involving alternating Euler sums which are defined by $$ {\zeta}(s_1,\dots,s_d;{\varepsilon}_1, \dots,{\varepsilon}_d)=\sum_{n_1>\cdots>n_d\ge1} \frac{{\varepsilon}_1^{n_1}\cdots{\varepsilon}_d^{n_d}}{ n_1^{s_1}\cdots n_d^{s_d}}, $$ for all positive integers s1,…,sd and e1=±1,…,ed=±1 with (s1,e1)≠(1,1). We call w=s1+⋯+sd the weight and d the depth. When ej=−1 we say the jth component is alternating. We first consider Euler sums of the following special type: $$ {\xi}(2s_1,\dots,2s_d)={\zeta}\bigl(2s_1, \dots,2s_d;(-1)^{s_1},\dots ,(-1)^{s_d}\bigr). $$ For d≤n, let Ξ(2n,d) be the sum of all ξ(2s1,…,2sd) of fixed weight 2n and depth d. We derive a formula for Ξ(2n,d) using the theory of symmetric functions established by Hoffman recently. We also consider restricted sum formulas of Euler sums with fixed weight 2n, depth d and fixed number α of alternating components at even arguments. When α=1 or α=d, we can determine precisely the restricted sum formulas. For other α we only treat the cases d<5 completely since the symmetric function theory becomes more and more unwieldy to work with when α moves closer to d/2.

A FAMILY OF MULTIPLE HARMONIC SUM AND MULTIPLE ZETA STAR VALUE IDENTITIES

In this paper we present a new family of identities for multiple harmonic sums which generalize a recent result of Hessami Pilehrood et al [Trans. Amer. Math. Soc. (to appear)]. We then apply it to obtain a family of identities relating multiple zeta star values to alternating Euler sums. In such a typical identity the entries of the multiple zeta star values consist of blocks of arbitrarily long 2-strings separated by positive integers greater than two while the largest depth of the alternating Euler sums depends only on the number of 2-string blocks but not on their lengths.

On the analytic computation of massless propagators in dimensional regularization

We comment on the algorithm to compute periods using hyperlogarithms, applied to massless Feynman integrals in the parametric representation. Explicitly, we give results for all three-loop propagators with arbitrary insertions including order ε4 and show examples at four and more loops.Further we prove that all coefficients of the ε-expansion of these integrals are rational linear combinations of multiple zeta values and in some cases possibly also alternating Euler sums.

ALTERNATING EULER SUMS AND SPECIAL VALUES OF THE WITTEN MULTIPLE ZETA FUNCTION ATTACHED TO

AbstractWe study the Witten multiple zeta function associated with the Lie algebra $\so $. Our main result shows that its special values at nonnegative integers are always expressible by alternating Euler sums. More precisely, every such special value of weight w at least 2 is a finite ℚ-linear combination of alternating Euler sums of weight w and depth at most 2, except when the only nonzero argument is one of the two last variables, in which case ζ(w−1) is needed.

Compact analytical form for non-zeta terms in critical exponents at order 1/ N3

We simplify, to a single integral of dilogarithms, the least tractable O(1/ N 3) contribution to the large- N critical exponent η of the non-linear σ-model, and hence φ 4-theory, for any spacetime dimensionality, D. It is the sole generator of irreducible multiple zeta values in ε-expansions with D=2−2 ε, for the σ-model, and D=4−2 ε, for φ 4-theory. In both cases we confirm results of Broadhurst, Gracey and Kreimer (BGK) that relate knots to counterterms. The new compact form is much simpler than that of BGK. It enables us to develop 8 new terms in the ε-expansion with D=3−2 ε. These involve alternating Euler sums, for which the basis of irreducibles is larger. We conclude that massless Feynman diagrams in odd spacetime dimensions share the greater transcendental complexity of massive diagrams in even dimensions, such as those contributing to the electron's magnetic moment and the electroweak ρ-parameter. Consequences for the perturbative sector of Chern-Simons theory are discussed.