Summation over k to n of Harmonic Number k by Harmonic Number n-k

Theorem

$\ds \sum_{k \mathop = 1}^n H_k H_{n - k} = \paren {n + 1} \paren { {H_n}^2 - H_n^{\paren 2} } - 2 n \paren {n_n - 1}$

where:

$H_n$ denotes the $n$th harmonic number

$H_n^{\paren 2}$ denotes a general harmonic number.

Proof

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Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $22$