Stieltjes

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1: 31.15 Stieltjes Polynomials

§31.15 Stieltjes Polynomials

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§31.15(ii) Zeros

… ►This is the Stieltjes electrostatic interpretation. … ►

§31.15(iii) Products of Stieltjes Polynomials

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2: 18.40 Methods of Computation

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§18.40(ii) The Classical Moment Problem

… ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain $w(x)$. ►

Stieltjes Inversion via (approximate) Analytic Continuation

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Histogram Approach

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Derivative Rule Approach

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3: 25.2 Definition and Expansions

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25.2.4 $$\zeta \left(s\right)=\frac{1}{s-1}+\sum _{n=0}^{\mathrm{\infty }}\frac{{(-1)}^n}{n!}{\gamma }_n{(s-1)}^n,$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, ${\gamma }_n$: Stieltjes constants, $!$: factorial (as in $n!$), $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series, Laurent series

Sources:

Hardy (1912, p. 216); mistakenly missing factor of ${(-1)}^n/n!$; Ivić (1985, (1.11), p. 4)

A&S Ref:

23.2.5 (with unnecessary constraint removed)

Referenced by:

Erratum (V1.0.17) for Equation (25.2.4), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E4

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

Originally this equation had the constraint $\mathrm{\Re }s>0$. This constraint is unnecessary because, as stated after (25.2.1), $\zeta \left(s\right)$ is meromorphic with a simple pole at $s=1$.

Suggested 2017-12-05 by John Harper

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

►where the Stieltjes constants ${\gamma }_n$ are defined via ►

25.2.5 $${\gamma }_n=\underset{m\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^m\frac{{(\ln k)}^n}{k}-\frac{{(\ln m)}^{n+1}}{n+1}\right).$$

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Defines:

${\gamma }_n$: Stieltjes constants

Symbols:

$\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

definition, Stieltjes constants

Sources:

Hardy (1912, p. 215); Ivić (1985, (1.12), p. 4)

Referenced by:

§25.2(ii), §25.6(ii), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

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4: 18.29 Asymptotic Approximations for $q$-Hahn and Askey–Wilson Classes

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18.29.2 $$Q_n(z;a,b,c,d\mid q)\sim \frac{z^n{(a{z}^{-1},b{z}^{-1},c{z}^{-1},d{z}^{-1};q)}_{\mathrm{\infty }}}{{(z^{-2},bc,bd,cd;q)}_{\mathrm{\infty }}},$$ $n\to \mathrm{\infty }$; $z,a,b,c,d,q$ fixed.

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Symbols:

$\sim$: asymptotic equality, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $z$: complex variable, $q$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.29.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.29 and Ch.18

►For a uniform asymptotic expansion of the Stieltjes–Wigert polynomials, see Wang and Wong (2006). …

5: 2.6 Distributional Methods

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§2.6(ii) Stieltjes Transform

… ►The Stieltjes transform of $f(t)$ is defined by … ►For a more detailed discussion of the derivation of asymptotic expansions of Stieltjes transforms by the distribution method, see McClure and Wong (1978) and Wong (1989, Chapter 6). 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). …

6: 1.4 Calculus of One Variable

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Stieltjes, Lebesgue, and Lebesgue–Stieltjes integrals

… ► … ► … ► … ► See Riesz and Sz.-Nagy (1990, Ch. 3). …

7: 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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8: 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). …

9: 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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10: 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:

pMML, png, TeX

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:

pMML, png, TeX

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) §18.27

Permalink:

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

Encodings:

pMML, png, TeX

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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