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
Famous quotes containing the words recurrence and/or formula:
“Forgetfulness is necessary to remembrance. Ideas are retained by renovation of that impression which time is always wearing away, and which new images are striving to obliterate. If useless thoughts could be expelled from the mind, all the valuable parts of our knowledge would more frequently recur, and every recurrence would reinstate them in their former place.”
—Samuel Johnson (1709–1784)
“Hidden away amongst Aschenbach’s writing was a passage directly asserting that nearly all the great things that exist owe their existence to a defiant despite: it is despite grief and anguish, despite poverty, loneliness, bodily weakness, vice and passion and a thousand inhibitions, that they have come into being at all. But this was more than an observation, it was an experience, it was positively the formula of his life and his fame, the key to his work.”
—Thomas Mann (18751955)