Stieltjes electrostatic interpretation

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11: 1.14 Integral Transforms

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§1.14(vi) Stieltjes Transform

►The Stieltjes transform of a real-valued function $f(t)$ is defined by … … ►

Inversion

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Laplace Transform

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12: 3.10 Continued Fractions

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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). …

13: 1.1 Special Notation

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$x,y$	real variables.
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$L^2(X,d\alpha )$	the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $d\alpha$.
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14: Tom H. Koornwinder

… ►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …

15: 18.27 $q$-Hahn Class

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§18.27(vi) Stieltjes–Wigert Polynomials

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18.27.19 $${\int }_0^{\mathrm{\infty }}\frac{S_n(x;q)S_m(x;q)}{{(-x,-q{x}^{-1};q)}_{\mathrm{\infty }}}dx=\frac{\ln \left(q^{-1}\right)}{q^n}\frac{{(q;q)}_{\mathrm{\infty }}}{{(q;q)}_n}{\delta }_{n,m},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $S_n(x;q)$: Stieltjes–Wigert polynomial, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.27.E19

Encodings:

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See also:

Annotations for §18.27(vi), §18.27 and Ch.18

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18.27.20 $${\int }_0^{\mathrm{\infty }}{S}_n(q^{\frac{1}{2}}x;q)S_m(q^{\frac{1}{2}}x;q)\exp \left(-\frac{{(\ln x)}^2}{2\ln \left(q^{-1}\right)}\right)dx=\frac{\sqrt{2\pi q^{-1}\ln \left(q^{-1}\right)}}{q^n{(q;q)}_n}{\delta }_{n,m}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $S_n(x;q)$: Stieltjes–Wigert polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $\ln z$: principal branch of logarithm function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.27.E20

Encodings:

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See also:

Annotations for §18.27(vi), §18.27 and Ch.18

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From Stieltjes–Wigert to Hermite

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18.27.20_5 $$\underset{q\to 1}{lim}\frac{{(q;q)}_n{S}_n(q^{-1}x\sqrt{2(1-q)}+1;q)}{{\left(\frac{1-q}{2}\right)}^{n/2}}={(-1)}^n{H}_n\left(x\right).$$

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Symbols:

$H_n\left(x\right)$: Hermite polynomial, $S_n(x;q)$: Stieltjes–Wigert polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.27.14))

Referenced by:

§18.27(vi), Erratum (V1.2.0) Section 18.27

Permalink:

http://dlmf.nist.gov/18.27.E20_5

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Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(vi), §18.27(vi), §18.27 and Ch.18

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16: Bibliography R

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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.

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Notes:

arXiv:1812.02196

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (4), pp. 91.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging. Computing in Science and Engineering 23 (3), pp. 56–64.

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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

Encodings:

BibTeX

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17: Bibliography I

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An electrostatics model for zeros of general orthogonal polynomials. Pacific J. Math. 193 (2), pp. 355–369.

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External Links:

ISSN 0030-8730, FullText, MathReview (T. Erber), ZentralBlatt

Referenced by:

§18.39(v).

Encodings:

BibTeX

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More on electrostatic models for zeros of orthogonal polynomials. Numer. Funct. Anal. Optim. 21 (1-2), pp. 191–204.

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External Links:

ISSN 0163-0563, FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§18.39(v).

Encodings:

BibTeX

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18: 9.17 Methods of Computation

… ►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. …

19: 25.6 Integer Arguments

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25.6.12 $${\zeta }^{\prime \prime }\left(0\right)=-\frac{1}{2}{(\ln \left(2\pi \right))}^2+\frac{1}{2}{\gamma }^2-\frac{1}{24}{\pi }^2+{\gamma }_1,$$

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $\pi$: the ratio of the circumference of a circle to its diameter and $\ln z$: principal branch of logarithm function

Source:

Apostol (1985a, pp. 226, 227, 229)

Referenced by:

Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.6.E12

Encodings:

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See also:

Annotations for §25.6(ii), §25.6 and Ch.25

►where ${\gamma }_1$ is given by (25.2.5). …

20: 28.16 Asymptotic Expansions for Large $q$

… ►For graphical interpretation, see Figures 28.13.1 and 28.13.2. …