the Digamma and Polygamma functions
Psi(x) is the digamma function,
Psi(n, x) is the nth polygamma function, which is the nth derivative of the digamma function when n is a nonnegative integer.
You can enter the command Psi using either the 1-D or 2-D calling sequence.
If n is an integer greater than one, Psi(n) + gamma is a rational number. (gamma is Euler's constant.) For small values of n, Psi(n) computes as a sum of gamma and a rational number. To perform this computation for larger values of n, use expand.
Psi(n, x) is extended to complex n, including negative integer indices, by the balanced polygamma formula of Espinosa and Moll
where ζ is the Hurwitz zeta function.
Evaluating Psi(51) directly is faster than expanding and then evaluating.
Unlike the negapolygamma of Gosper, the balanced polygamma at n=−1 differs from lnGAMMA by a constant
Espinosa, O., and Moll, V. "A Generalized Polygamma Function." Integral Transforms and Special Functions, (April 2004): 101-115.
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