# Stieltjes electrostatic interpretation

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##### 1: 31.15 Stieltjes Polynomials
###### §31.15(ii) Zeros
This is the Stieltjes electrostatic interpretation. …
##### 2: 2.6 Distributional Methods
With this interpretation
###### §2.6(ii) Stieltjes Transform
The Stieltjes transform of $f(t)$ is defined by … Corresponding results for the generalized Stieltjes transform …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). …
##### 3: 18.39 Physical Applications
For physical applications of $q$-Laguerre polynomials see §17.17. For interpretations of zeros of classical OP’s as equilibrium positions of charges in electrostatic problems (assuming logarithmic interaction), see Ismail (2000a, b).
##### 4: 29.12 Definitions
29.12.13 ${\frac{\rho+\frac{1}{4}}{\xi_{p}}+\frac{\sigma+\frac{1}{4}}{\xi_{p}-1}+\frac{% \tau+\frac{1}{4}}{\xi_{p}-k^{-2}}+\sum_{\begin{subarray}{c}q=1\\ q\neq p\end{subarray}}^{n}\frac{1}{\xi_{p}-\xi_{q}}=0},$ $p=1,2,\dots,n$.
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 $j=1,2,\dots,n$.
##### 5: 25.2 Definition and Expansions
25.2.4 $\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma% _{n}(s-1)^{n},$
where the Stieltjes constants $\gamma_{n}$ are defined via
25.2.5 $\gamma_{n}=\lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln k)^{n}}{k}-\frac{(% \ln m)^{n+1}}{n+1}\right).$
##### 6: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes
18.29.2 $Q_{n}(z;a,b,c,d\mid q)\sim\frac{z^{n}\left(az^{-1},bz^{-1},cz^{-1},dz^{-1};q% \right)_{\infty}}{\left(z^{-2},bc,bd,cd;q\right)_{\infty}},$ $n\to\infty$; $z,a,b,c,d,q$ fixed.
For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). …
##### 7: Bibliography K
• A. V. Kashevarov (2004) The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface. Technical Physics 49 (1), pp. 1–7.
• T. H. Koornwinder (2015) Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.
• ##### 8: 1.14 Integral Transforms
###### §1.14(vi) Stieltjes Transform
The Stieltjes transform of a real-valued function $f(t)$ is defined by … …
##### 9: 3.10 Continued Fractions
###### Stieltjes Fractions
is called a Stieltjes fraction ($S$-fraction). … For the same function $f(z)$, the convergent $C_{n}$ of the Jacobi fraction (3.10.11) equals the convergent $C_{2n}$ of the Stieltjes fraction (3.10.6). …
##### 10: 18.27 $q$-Hahn Class
###### §18.27(vi) Stieltjes–Wigert Polynomials
18.27.18 $S_{n}\left(x;q\right)=\sum_{\ell=0}^{n}\frac{q^{\ell^{2}}(-x)^{\ell}}{\left(q;% q\right)_{\ell}\left(q;q\right)_{n-\ell}}=\frac{1}{\left(q;q\right)_{n}}{{}_{1% }\phi_{1}}\left({q^{-n}\atop 0};q,-q^{n+1}x\right).$
18.27.19 $\int_{0}^{\infty}\frac{S_{n}\left(x;q\right)S_{m}\left(x;q\right)}{\left(-x,-% qx^{-1};q\right)_{\infty}}\mathrm{d}x=\frac{\ln\left(q^{-1}\right)}{q^{n}}% \frac{\left(q;q\right)_{\infty}}{\left(q;q\right)_{n}}\delta_{n,m},$
18.27.20 $\int_{0}^{\infty}S_{n}\left(q^{\frac{1}{2}}x;q\right)S_{m}\left(q^{\frac{1}{2}% }x;q\right)\exp\left(-\frac{(\ln x)^{2}}{2\ln\left(q^{-1}\right)}\right)% \mathrm{d}x=\frac{\sqrt{2\pi q^{-1}\ln\left(q^{-1}\right)}}{q^{n}\left(q;q% \right)_{n}}\delta_{n,m}.$