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A generalized polygamma function

Olivier Espinosa·Victor H. Moll
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Abstract

We study the properties of a function $ψ(z, q)$ (the generalized polygamma function), intimately connected with the Hurwitz zeta function and defined for complex values of the variables $z$ and $q$, which is entire in the variable $z$ and reduces to the usual polygamma function $ψ^{(m)}(q)$ for $z$ a non-negative integer $m$, and to the balanced negapolygamma function $ψ^{(-m)}(q)$ for $z$ a negative integer $-m$.

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Olivier Espinosa, Victor H. Moll. 2003-05-05. A generalized polygamma function. https://arxiv.org/abs/math/0305079

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