# Digamma at z+n

From specialfunctionswiki

## 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.

## Proof

## References

- 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi:
*Higher Transcendental Functions Volume I*... (previous) ... (next): $\S 1.7 (10)$