The difference between two digamma function can be written using the following recurrence relation:
$\psi(n+z) - \psi(z) = \sum_{i=0}^{n} \frac{1}{i + z}$
My question is, is there a closed form solution to this recurrence function? and how can i get?
I appologize my question was not clear. I want a 'function' to compute the right hand side efficiently $\sum_{i=0}^{n-1} \frac{1}{i + z}$, without the need to do linear increment if possible and also without using the digamma function. I do not want to use the digamma function because it is expensive. I used this recurrence relation for the difference of the two digamma function in the first place to avoid evaluating $\psi(x)$.