Definition:Harmonic Numbers/Notation
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Harmonic Numbers: Notation
There appears to be no standard notation for the harmonic numbers.
The notation given here, and used on $\mathsf{Pr} \infty \mathsf{fWiki}$ throughout, is an adaptation for $\mathsf{Pr} \infty \mathsf{fWiki}$ of an idea by Donald E. Knuth, where he used $H_n^{\paren r}$.
Knuth's notation proves unwieldy when extended to the complex numbers, and so we have adopted the more conventional mapping notation $\harm r n$ and hence $\harm r z$.
Ramanujan used $\ds \map {\phi_r} n$.
Other notations that can also be found in the literature include $h_n$, $S_n$ and $\map \psi {n + 1} + \gamma$.
Sources
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- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: $(1)$
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): harmonic number