Minking Eie

ON GENERALIZATIONS OF WEIGHTED SUM FORMULAS OF MULTIPLE ZETA VALUES

In this paper, we compute shuffle relations from multiple zeta values of the form ζ({1}m-1, n+1) or sums of multiple zeta values of fixed weight and depth. Some interesting weighted sum formulas are obtained, such as where m and k are positive integers with m ≥ 2k. For k = 1, this gives Ohno–Zudilins weighted sum formula.

A theorem on zeta functions associated with polynomials

Let β = (β1, . . . , βr) be an r-tuple of non-negative integers and Pj(X) (j = 1, 2, . . . , n) be polynomials in R[X1, . . . , Xr ] such that Pj(n) > 0 for all n ∈ Nr and the series ∑ n∈Nr Pj(n) −s is absolutely convergent for Re s > σj > 0. We consider the zeta functions Z(Pj , β, s) = ∑ n∈Nr nPj(n) −s, Re s > |β|+ σj , 1 ≤ j ≤ n. All these zeta functions Z( ∏n j=1 Pj , β, s) and Z(Pj , β, s) (j = 1, 2, . . . , n) are analytic functions of s when Re s is sufficiently large and they have meromorphic analytic continuations in the whole complex plane. In this paper we shall prove that Z( n ∏ j=1 Pj , β, 0) = 1 n n ∑ j=1 Z(Pj , β, 0). As an immediate application, we use it to evaluate the special values of zeta functions associated with products of linear forms as considered by Shintani and the first author.

EXPLICIT EVALUATION OF TRIPLE EULER SUMS

The triple Euler sum defined by \[ \begin{array}{rcl} \zeta(p, q, r) & = & \sum\limits_{\ell=3}^{\infty} \frac{1}{\ell^{p}} \sum\limits_{k=2}^{\ell-1} \frac{1}{k^{q}} \sum\limits_{j=1}^{k-1} \frac{1}{j^{r}} \\ [10pt] & = & \sum\limits_{l=1}^{\infty} \sum\limits_{k=1}^{\infty} \sum\limits_{j=1}^{\infty} \frac{1}{(j+k+\ell)^{p}(j+k)^{q}j^{r}} \end{array} \] with positive integers p, q, r, p ≥ 2, has not been fully evaluated yet except for the case of q = r = 1 and some particular cases such as p + q + r ≤ 10 or p = q = r. With the general theory developed for double Euler sums, we are able to produce identities among triple Euler sums with a variable or two when q = r =. Performing differentiations with respect to the specified variable gives the explicit evaluations of ζ(p, q, r) for general p, q, r. In particular, the values of ζ(n, 1, 1), ζ(2n + 1, 1, 2), ζ(2n + 1, 2, 1) and ζ(2n, 2, 2) are given explicitly in terms of classical double Euler sums. Also ζ(2n, 1, 2m + 1) and ζ(2n + 1, 1, 2m) are obtained f...

ON EVALUATION OF GENERALIZED EULER SUMS OF EVEN WEIGHT

The classical Euler sum cannot be evaluated when the weight p + q is even unless p = 1 or p = q or (p, q) = (2, 4) or (p, q) = (4, 2) [7]. However it is a different story if instead we consider the alternating sums and They can be evaluated for even weight p + q. In this paper, we shall evaluate a family of generalized Euler sums containing when the weight p + q is even via integral transforms of Bernoulli identities.

A generalization of rummer’s congruences

AbstractSuppose thatm, n are positive even integers andp is a prime number such thatp-1 is not a divisor ofm. For any non-negative integerN, the classical Kummer’s congruences on Bernoulli numbersBn(n = 1,2,3,...) assert that (1-pm-1)Bm/m isp-integral and(1)

Several weighted sum formulas of multiple zeta values

(1 - p^{m - 1} )\frac{{B_m }}{m} \equiv (1 - p^{n - 1} )\frac{{B_n }}{n}(\bmod p^{N + 1} )

Some alternating double sum formulae of multiple zeta values

ifm ≡ n (mod (p-1)pn). In this paper, we shall prove that for any positive integerk relatively prime top and non-negative integers α, β such that α +jk =pβ for some integerj with 0 ≤j ≤p-l.Then for any non-negative integerN,(2)