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I asked the same question on here but received no answer.

The classic problem of the electrostatic equilibrium positions of a linear system of $ n $ free unit charges between two fixed charges is well known, here an example with $n = 1$.

Few days ago I found out a nice result going back to Stieltjes and Hasse: if the potential between the two fixed charges is logarithmic, then the positions mentioned above can be computed as the roots of the Jacobi polynomial $P_{n}^{(2p-1,2q-1)}$, where $p$ is the fixed charge positioned in position $-1$, $q$ is is the fixed charge in position $+1$ and $n$ is the number of free unit charges (I attached an extract from Szego's book "Orthogonal polynomials").

My question is if there exist anything similar for the Coulomb potential $V=1/r$. I found some reference at the end of this paper, but I would know more.

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  • $\begingroup$ Where did you ask the same question? Your link goes to a picture. $\endgroup$
    – Gerry Myerson
    Commented May 21, 2023 at 0:23
  • $\begingroup$ @GerryMyerson sorry, just edited $\endgroup$
    – user967210
    Commented May 21, 2023 at 7:37
  • $\begingroup$ Not entirely sure how related this is: there is an extension of the Stieltjes result to the six classical orthogonal polynomials that has a 1/r interaction between roots (arxiv.org/pdf/1804.09697.pdf). The induced energy can be used to quickly find roots of the six classical orthogonal polynomials by solving a coupled system of ODEs (faster methods exist, of course). $\endgroup$
    – Stefan Steinerberger
    Commented Jun 5, 2023 at 22:10

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