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Stieltjes electrostatic interpretation

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1: 31.15 Stieltjes Polynomials
§31.15 Stieltjes Polynomials
§31.15(ii) Zeros
This is the Stieltjes electrostatic interpretation. …
§31.15(iii) Products of Stieltjes Polynomials
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 ρ + 1 4 ξ p + σ + 1 4 ξ p - 1 + τ + 1 4 ξ p - k - 2 + q = 1 q p n 1 ξ p - ξ q = 0 , p = 1 , 2 , , 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 ρ + 1 4 , σ + 1 4 , and τ + 1 4 , respectively, and n movable point masses at t 1 , t 2 , , t n arranged according to (29.12.12) with unit positive charges, the equilibrium position is attained when t j = ξ j for j = 1 , 2 , , n .
5: 25.2 Definition and Expansions
25.2.4 ζ ( s ) = 1 s - 1 + n = 0 ( - 1 ) n n ! γ n ( s - 1 ) n ,
where the Stieltjes constants γ n are defined via
25.2.5 γ n = lim m ( k = 1 m ( ln k ) n k - ( ln m ) n + 1 n + 1 ) .
6: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
18.29.2 Q n ( z ; a , b , c , d q ) z n ( a z - 1 , b z - 1 , c z - 1 , d z - 1 ; q ) ( z - 2 , b c , b d , c d ; q ) , n ; 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 … …
    Inversion
    Laplace Transform
    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 2 n of the Stieltjes fraction (3.10.6). …
    10: 18.27 q -Hahn Class
    §18.27(vi) Stieltjes–Wigert Polynomials
    18.27.18 S n ( x ; q ) = = 0 n q 2 ( - x ) ( q ; q ) ( q ; q ) n - = 1 ( q ; q ) n ϕ 1 1 ( q - n 0 ; q , - q n + 1 x ) .