electrostatic interpretation

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1: 31.15 Stieltjes Polynomials

… ►This is the Stieltjes electrostatic interpretation. …

2: 29.12 Definitions

… ►

29.12.13

{\frac{\rho+\frac{1}{4}}{\xi_{p}}+\frac{\sigma+\frac{1}{4}}{\xi_{p}-1}+\frac{% \tau+\frac{1}{4}}{\xi_{p}-k^{-2}}+\sum_{\begin{subarray}{c}q=1\\ q\neq p\end{subarray}}^{n}\frac{1}{\xi_{p}-\xi_{q}}=0},

p=1,2,\dots,n

ⓘ

Symbols:: $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : real parameter, ( $0$ or $\tfrac{1}{2}$ ): parameter $\rho$ , ( $0$ or $\tfrac{1}{2}$ ): parameter $\sigma$ , ( $0$ or $\tfrac{1}{2}$ ): parameter $\tau$ and $\xi_{n}$ : zeros
Referenced by:: §29.20(iii)
Permalink:: http://dlmf.nist.gov/29.12.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §29.12(iii), §29.12 and Ch.29

►This result admits the following electrostatic interpretation: Given three point masses fixed at

t=0

t=1

, and

t=k^{-2}

with positive charges

\rho+\tfrac{1}{4}

\sigma+\tfrac{1}{4}

, and

\tau+\tfrac{1}{4}

, respectively, and

n

movable point masses at

t_{1},t_{2},\dots,t_{n}

arranged according to (29.12.12) with unit positive charges, the equilibrium position is attained when

t_{j}=\xi_{j}

for

j=1,2,\dots,n

3: 18.39 Applications in the Physical Sciences

… ►For interpretations of zeros of classical OP’s as equilibrium positions of charges in electrostatic problems (assuming logarithmic interaction), see Ismail (2000a, b).