Recurrence Relation
It satisfies the recurrence relation
which – considered for positive integer argument – leads to a presentation of the sum of reciprocals of the powers of the natural numbers:
and
for all . Like the -function, the polygamma functions can be generalized from the domain uniquely to positive real numbers only due to their recurrence relation and one given function-value, say, except in the case m=0 where the additional condition of strictly monotony on is still needed. This is a trivial consequence of the Bohr–Mollerup theorem for the gamma function where strictly logarithmic convexity on is demanded additionaly. The case m=0 must be treated differently because is not normalizable at infinity (the sum of the reciprocals doesn't converge).
Read more about this topic: Polygamma Function
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