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Are there situations in which the polygamma pops up naturally in a mathematical physics context? In particular: are there examples of potentials having some interest for which the dependence on the distance is expressed in terms of $\psi^{(n)}$?

Update: While Carlo Beenakker's answer is clearly useful, the references therein still don't contain exactly what I'm looking for in the second part of the question.

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2 Answers 2

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Q: Are there situations in which the polygamma function pops up naturally in a mathematical physics context?

A: Certainly, for example:

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Abou-Salem, L. I., A study on baryons spectroscopy using digamma-function as interacting potential, https://arxiv.org/abs/1311.6743 studies using the digamma function as an interaction potential (for quarks in baryons).

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