EVALUATION OF EULER-ZAGIER SUMS

We present a simple method for evaluation of multiple Euler sums in terms of single and double zeta values. 2000 Mathematics Subject Classification. 11M99, 40B05.

Multiple Euler sums have been discussed and evaluated in a number of papers of which we want to point out [1,2,3,4,5,6,7,10].Also [8,Sections 18 and 19].We refer to these publications for general comments and details.
Proof.We have (2.2) Equation (2.1) follows by repeating the procedure p − 1 times in view of the fact that (see [9, page 665]) Now we differentiate (2.1) r − 1 times, where r > 1.With D = d/dx we have (2.4) Therefore, x r+p−k−1 . ( We summarize this result in the following lemma.
By setting x = 1 we get the desired representation of w(p, q, r ).Making use of (see [9, page 775]), and with the agreement to read ζ(1) = 0, one obtains the following corollary.
Corollary 2.4.For all integers p > 1, r ≥ 1 and all q ≥ 0 with q + r > 1, (2.12) When q > 0 (or q ≥ 1, p = 1) we also have (2.13) 3. Remarks.Our notation S(p, q) corresponds to S p,q in [5].The authors of [2] use the sums ζ(p, q), which equal S(q, p) The representation (2.11) has strong and weak points.One good feature is that q need not be an integer.A weak point is that the right-hand side in (2.11) is not explicitly symmetrical in p and q, while obviously w(p, q, r ) = w(q, p, r ).Moreover, the right-hand side has too many terms.For instance, when q = 0 (2.11) becomes (here the term (−1) p−1 S(r , p) is written separately on purpose).At the same time which is much shorter.However, we can benefit from this situation if we compare the two representations of w(p, 0,r ) and derive relations for the single and double Euler sums.For instance, when p is odd, we can solve for S(r , p) to get that is, S(r , p) can be expressed in terms of single zeta values and S(k, l), with k < r , k + l = r + p.

Other sums.
It is interesting to consider also the sum and compare it to w(p, q, r ).Here one can write Let q >p.We observe that if (q−p)/r is odd, repeating this step (q−p)/r times, we get u(p,q,r ) = from where, because of the symmetry u(p,q,r ) = u(q,p,r ), we obtain the following proposition.