Balanced polygamma function
It generalizes the polygamma function to negative and fractional order, but remains equal to it for integer positive orders.
The generalized polygamma function is defined as follows:
The function is balanced, in that it satisfies the conditions and .
Several special functions can be expressed in terms of generalized polygamma function.
where are Bernoulli polynomials
The balanced polygamma function can be expressed in a closed form at certain points: