On Spieß’s conjecture on harmonic numbers

Let HnHn be the nn-th harmonic number and let Hn(2) be the nn-th generalized harmonic number of order two. Spieß proved that for a nonnegative integer mm and for t=1,2t=1,2, and 33, the sum R(m,t)=∑k=0nkmHkt can be represented as a polynomial in HnHn with polynomial coefficients in nn plus Hn(2) multiplied by a polynomial in nn. For t=3t=3, we show that the coefficient of Hn(2) in Spieß’s formula equals Bm/2Bm/2, where BmBm is the mm-th Bernoulli number. Spieß further conjectured for t≥4t≥4 such a summation takes the same form as for t≤3t≤3. We find a counterexample for t=4t=4. However, we prove that the structure theorem of Spieß holds for the sum ∑k=0np(k)Hk4 when the polynomial p(k)p(k) satisfies a certain condition. We also give a structure theorem for the sum ∑k=0nkmHkHk(2). Our proofs rely on Abel’s lemma on summation by parts.