Digamma Function - Relation To Harmonic Numbers

Relation To Harmonic Numbers

The digamma function, often denoted also as ψ0(x), ψ0(x) or (after the shape of the archaic Greek letter Ϝ digamma), is related to the harmonic numbers in that

where Hn is the n harmonic number, and γ is the Euler-Mascheroni constant. For half-integer values, it may be expressed as

\psi\left(n+{\frac{1}{2}}\right) = -\gamma - 2\ln 2 + \sum_{k=1}^n \frac{2}{2k-1}

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