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Generalized reciprocity laws for sums of harmonic numbers. (English) Zbl 1202.68492
Summary: We present summation identities for generalized harmonic numbers \(H_ n^ {(a)}=\sum_ {k=1}^ n \frac{1}{k^ a}\), which generalize reciprocity laws discovered when studying the algorithm quickselect [P. Kirschenhofer and H. Prodinger, Comb. Probab. Comput. 7, No. 1, 111–120 (1998; Zbl 0892.68021)]. Furthermore, we demonstrate how the computer algebra package Sigma can be used in order to find/prove such identities. We also discuss alternating harmonic sums, as well as limiting relations.

MSC:
68W30 Symbolic computation and algebraic computation
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
68W40 Analysis of algorithms
11B50 Sequences (mod \(m\))
Software:
OEIS; SIGMA
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