## The Many Faces of the Polygamma Function

**Authors:** Simon Plouffe

A survey is made based on finite sums of the polygamma function with rational arguments which are D_(k,j)^n=∑_((m,n)=1)▒〖χ_j (n)ψ(k,m/n) 〗 Where, χ_j (n) is the j’th Dirichlet character and ψ(k,m/n) is the polygamma function of order k. We use this representation to rewrite identities using a new notation for linear combinations of mathematical constants. Identities are given for prime numbers using irrational constants. For negative argument n we use the generalization of Espinosa and Moll[6], well implemented into Maple CAS.

**Comments:** A minor correction on page 2

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### Submission history

[v1] 2016-03-31 22:44:34

[v2] 2018-08-13 13:46:02

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