Explicit formulas of Euler sums via multiple zeta values

Shuffle Product Formulas of Multiple Zeta Values

Using the combinatorial description of shuffle product, we prove or reformulate several shuffle product formulas of multiple zeta values, including a general formula of the shuffle product of two multiple zeta values, some restricted shuffle product formulas of the product of two multiple zeta values, and a restricted shuffle product formula of the product of n multiple zeta values.

ALTERNATING EULER SUMS AND SPECIAL VALUES OF WITTEN MULTIPLE ZETA FUNCTION ATTACHED TO so(5)

Abstract. In this note we shall study the Witten multiple zeta function associated to the Lie algebra so(5) defined by Matsumoto. 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 ≥ 3 is a finite rational linear combination of alternating Euler sums of weight w and depth at...

Central Binomial Sums, Multiple Clausen Values, and Zeta Values

We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apéry sums). The study of non-alternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ra...

Explicit evaluation of Euler sums

In response to a letter from Goldbach, Euler considered sums of the form where s and t are positive integers. As Euler discovered by a process of extrapolation (from s + t 13), h (s; t) can be evaluated in terms of Riemann-functions when s + t is odd. We provide a rigorous proof of Euler's discovery and then give analogous evaluations with proofs for corresponding alternating sums. Relatedly we...

New Families of Weighted Sum Formulas for Multiple Zeta Values