Euler Sums Research Articles

POLYLOGARITHMIC CONNECTIONS WITH EULER SUMS

POLYLOGARITHMIC CONNECTIONS WITH EULER SUMS

Finite Mixed Sums wih Harmonic Terms

Abstract In the spirit of Euler sums we develop a set of identities for finite sums of products of harmonic numbers in higher order and reciprocal binomial coefficients. The new results complement some Euler sums of the type

Sum formulas and duality theorems of multiple zeta values

Multiple zeta values or r-ford Euler sums are defined byζ(α1,α2,…,αr)=∑1≤n1<n2<⋯<nrn1−α1n2−α2⋯nr−αr with α1,α2,…,αr are positive integers and αr≥2. Let |α|=α1+α2+⋯+αr, the sum formula then asserted that∑|α|=mζ(α1,α2,…,αr+1)=ζ(m+1) for any positive integer m≥r.In this paper, we investigate the multiple zeta values with a parameter a>−1 given by∑1≤n1<n2<⋯<np+r1n1n2⋯np(np+1+a)⋯(np+r+a)np+r as well as their vector versions. Among other things, we obtain that the above multiple zeta values are equal to∑|α|=p+r∑1≤k1<k2<⋯<kp+1k1−α1k2−α2⋯kp+1−αp+1(kp+1+a)−1. By differentiations with respect to a and then set a=0, we obtain some interesting sum formulas including the aforementioned sum formula and the restricted sum formula∑|α|=mζ({1}p,α1,…,αr+1)=∑|c|=p+rζ(c1,c2,…,cp+1+(m−r)+1). Also, a generalized duality theorem is given in the final section.

Families of weighted sum formulas for multiple zeta values

Euler's sum formula and its multi-variable and weighted generalizations form a large class of the identities of multiple zeta values. In this paper, we prove a family of identities involving Bernoulli numbers and apply them to obtain infinitely many weighted sum formulas for double zeta values and triple zeta values where the weight coefficients are given by symmetric polynomials. We give a general conjecture in arbitrary depth at the end of the paper.

Sums of Involving the Harmonic Numbers and the Binomial Coefficients

Let the numbers be defined by , where and are the exponential complete Bell polynomials. In this paper, by means of the methods of Riordan arrays, we establish general identities involving the numbers , binomial coefficients and inverse of binomial coefficients. From these identities, we deduce some identities involving binomial coefficients, Harmonic numbers and the Euler sum identities. Furthermore, we obtain the asymptotic values of some summations associated with the numbers by Darboux’s method.

Weighted sum formulas from shuffle products of multiples of Riemann zeta values

The classical Euler decomposition theorem expressed a product of two Riemann zeta values in terms of double Euler sums. It can also be obtained from the shuffle product of two Riemann zeta values when both are expressed as iterated integrals due to Kontsevich. In this paper, we investigate the shuffle product of n multiples of Riemann zeta values(cj+1)ζ(cj+2),j=1,2,…,n,cj≥0 through their particular integral representations. Among other things, we obtain the following weighted sum formula∑|α|=k∑βζ(α1+β1+1,α2+β2+1,…,αn+βn+1)∏j=1n(αj+βj)!αj!jαj=1n!∑|c|=k∏j=1n(cj+1)ζ(cj+2) with∑βx1β1x2β2⋯xnβn=(x1+x2+⋯+xn)(x2+⋯+xn)⋯xn which is a sum of n! monomials of degree n. Here ζ(a1,a2,…,an) appearing in the summation is a multiple zeta value or n-fold Euler sum. Also we obtain the weighted sum formula of triple Euler sums∑|α|=k+5ζ(α1,α2,α3+1)⋅{2α2−1(3α3−2α3)−(2α3−1)}−∑|α|=k+4ζ(1,α2,α3+1)⋅(2α3−1)+ζ(1,1,k+4)=16∑|c|=kζ(c1+2)ζ(c2+2)ζ(c3+2).

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.

Euler sums of hyperharmonic numbers

The hyperharmonic numbers hn(r) are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers:σ(r,m)=∑n=1∞hn(r)nm can be expressed in terms of series of Hurwitz zeta function values. This is a generalization of a result of Mező and Dil (2010) [7]. We also provide an explicit evaluation of σ(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.

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.

Symmetries of Bernoulli polynomial series and Arakawa–Kaneko zeta functions

TextThe Arakawa–Kaneko zeta functions interpolate the poly-Bernoulli numbers at the negative integers and their values at positive integers are connected to multiple zeta values. We give everywhere-convergent series expansions of these functions in terms of Bernoulli polynomials and Dirichlet series related to harmonic numbers, exhibiting their explicit analytic continuation. Differentiating the Barnes multiple zeta functions of order r with respect to their order produces Dirichlet series attached to Bernoulli polynomials. These series are invariant under an involution in which the order of the derivative is dual to the value of the first variable and the order of the zeta function is dual to the value of the second variable. This symmetry relation generalizes duality relations of Euler sums, and is featured in our series expansions of Arakawa–Kaneko zeta values. VideoFor a video summary of this paper, please visit http://youtu.be/0vQqfgrkX2k.

On representation problems of Euler sums with multi-parameters

In this paper we consider the following Euler sumswhere p and q are positive integers, z and zi(i=1, 2, … ) are complex numbers on the unit circle or points within the unit circle except the origin and x is 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.

On q-analogues of double Euler sums

In this paper we study properties of a q-analogue of the function π/sinπz which is defined by means of Jacksonʼs q-gamma function and a reflection formula for the gamma function. As application, we obtain evaluations for q-analogues of double Euler sums in terms of single q-zeta values. We also derive a q-analogue of a well-known formula for the multiple zeta-star value ζ⁎({2}a).

Some identities on the Hurwitz zeta function and the extended Euler sums

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

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.

Generalizations of Euler decomposition and their applications

The classical Euler decomposition theorem expressed a product of two Riemann zeta values in terms of double Euler sums. Such kind of decomposition theorem are useful to produce weighted sum formulae on double Euler sums. In this paper, we produce a more general decomposition theorem as a generalization of Euler decomposition theorem. Indeed, we obtained the shuffle product formula of the product ζ ( { 1 } j , m + r − ℓ + 2 ) ζ ( { 1 } k − j , q − m + ℓ + 2 ) with 0 ⩽ j ⩽ k , 0 ⩽ m ⩽ q and 0 ⩽ ℓ ⩽ r . Along with some well-known theorems from the theory of probability, we are able to build up a lot of double weighted sum formulae which express sums of multiple zeta values in terms of single zeta values. Such as ∑ c + d = k ∑ m = 0 q ∑ n = 0 r ∑ | α | = c + m + 1 , | β | = d + q − m + r + 2 , β d + 1 = q − m + n + 1 ζ ( α 0 , … , α c + β 0 , … , β d , β d + 1 + 1 ) ( β d + 1 − 1 n ) ( − 1 ) m + ∑ a + b = q ∑ | α | = k + q + r + 3 , α k + 1 = r + b + 2 ζ ( α 0 , … , α k , α k + 1 + 1 ) ( r + b r ) { ( − 1 ) a + ( − 1 ) q + r } = 1 2 ∑ j = 0 k ∑ m = 0 q ∑ ℓ = 0 r ( − 1 ) j + m + ℓ ζ ( { 1 } j , m + r − ℓ + 2 ) ζ ( { 1 } k − j , q − m + ℓ + 2 ) , where k + q + r is even.

Remarks on double zeta values of level 2

We give a generator of the space spanned by double zeta values of level 2 with odd weight by using explicit formulas for double Euler sums.

Evaluations of some quadruple Euler sums of even weight

For positive integers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r}$ with $\alpha_{r} \geq 2$, the multiple zeta value or $r$-fold Euler sum is defined by \[\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r}) = \sum_{1\leq n_{1}<n_{2}<\cdots<n_{r}}n_1^{-\alpha_1}n_2^{-\alpha_2}\cdots n_r^{-\alpha_r}. \] where $|\alpha|=\alpha_1+\alpha_2+\cdots+\alpha_r$ and $r$ are the weight and depth of $\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r})$ respectively. By the general theorem given in [11], the multiple zeta value $\zeta(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{r})$ can be expressed as a rational linear combination of products of multiple zeta values of lower depth if its depth and weight are of different parity. In other words, when the sum of its depth and weight is odd. However, there are still some exceptions for quadruple Euler sums. As conjectured in [6], a quadruple Euler sum with even weight exceeding 14 can be expressed as a rational linear combination of products of multiple zeta values of depth 1, 2 and 3 if and only if it is one of the following forms: $\zeta(1,a,b,a)$, $\zeta(b,1,a,a)$, $\zeta(b,b,1,a)$ or $\zeta(a,b,b,a)$ with $a=b$ or $b=1$. In this paper, we shall evaluate these quadruple Euler sums of even weight by the identities among multiple zeta values with variables and relation obtained from the stuffle formula of two multiple zeta values.

Harmonic number identities via hypergeometric series and Bell polynomials

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.

Riemann and Euler Sum Investigation in an Introductory Calculus Class

This paper provides a detailed outline of a mathematical research exploration for use in an introductory high school or college Calculus class and is directed toward teachers of such courses. The discovery is accomplished by introducing a novel method to generate a polynomial expression for each of the Euler sums, ΣNk=0kn,n∈Z+ . The described method flows simply from initial discussions of the Riemann sum definition of a definite integral and is readily accessible to all new calculus students. Students investigate the Bernoulli numbers and the interesting connections with Pascal's Triangle. Advice is offered throughout as to how the project can be assigned to students and offers multiple suggestions for additional exploration for any motivated student.

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.