Digamma at z+n

Theorem

The following formula holds for $n=1,2,3,\ldots$: $$\psi(z+n)=\dfrac{1}{z} + \dfrac{1}{z+1} + \ldots + \dfrac{1}{z+n-1} + \psi(z),$$ where $\psi$ denotes the digamma function.

References

• 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.7 (10)$