Recurrence Formula and Characterization
The digamma function satisfies the recurrence relation
- (see proof)
Thus, it can be said to "telescope" 1/x, for one has
where Δ is the forward difference operator. This satisfies the recurrence relation of a partial sum of the harmonic series, thus implying the formula
where is the Euler-Mascheroni constant.
More generally, one has
Actually, is the only solution of the functional equation that is monotone on and satisfies . This fact follows immediately from the uniqueness of the function given its recurrence equation and convexity-restriction.
Read more about this topic: Digamma Function
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