Concrete Mathematics: A Foundation for Computer Science
Concrete Mathematics: A Foundation for Computer Science
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Let H"n be the n-th harmonic number and let H"n^(^2^) be the n-th generalized harmonic number of order two. Spiesz proved that for a nonnegative integer m and for t=1,2, and 3, the sum R(m,t)=@?"k"="0^nk^mH"k^t can be represented as a polynomial in H"n with polynomial coefficients in n plus H"n^(^2^) multiplied by a polynomial in n. For t=3, we show that the coefficient of H"n^(^2^) in Spiesz's formula equals B"m/2, where B"m is the m-th Bernoulli number. Spiesz further conjectured for t=4 such a summation takes the same form as for t@?3. We find a counterexample for t=4. However, we prove that the structure theorem of Spiesz holds for the sum @?"k"="0^np(k)H"k^4 when the polynomial p(k) satisfies a certain condition. We also give a structure theorem for the sum @?"k"="0^nk^mH"kH"k^(^2^). Our proofs rely on Abel's lemma on summation by parts.