Stieltjes polynomials

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1: 31.15 Stieltjes Polynomials

§31.15 Stieltjes Polynomials

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§31.15(i) Definitions

►Stieltjes polynomials are polynomial solutions of the Fuchsian equation (31.14.1). … ►

§31.15(ii) Zeros

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§31.15(iii) Products of Stieltjes Polynomials

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2: 18.27 $q$-Hahn Class

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§18.27(vi) Stieltjes–Wigert Polynomials

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18.27.18 $$S_n(x;q)=\sum _{\mathrm{\ell }=0}^n\frac{q^{{\mathrm{\ell }}^2}{(-x)}^{\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}{(q;q)}_{n-\mathrm{\ell }}}=\frac{1}{{(q;q)}_n}{}_1\varphi _1(\genfrac{}{}{0pt}{}{q^{-n}}{0};q,-q^{n+1}x).$$

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Defines:

$S_n(x;q)$: Stieltjes–Wigert polynomial

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}\varphi _s(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}\varphi _s(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.27.E18

Encodings:

pMML, png, TeX

See also:

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

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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, $\int$: 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, $\exp z$: 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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3: 18.1 Notation

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Stieltjes–Wigert: $S_n(x;q)$.

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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: Bibliography V

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Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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External Links:

ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

Encodings:

BibTeX

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6: Bibliography W

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Uniform asymptotics of the Stieltjes-Wigert polynomials via the Riemann-Hilbert approach. J. Math. Pures Appl. (9) 85 (5), pp. 698–718.

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External Links:

ISSN 0021-7824, Document, MathReview (Rabah Khaldi), ZentralBlatt

Referenced by:

§18.29.

Encodings:

BibTeX

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7: Bibliography

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Zeros of Stieltjes and Van Vleck polynomials and applications. J. Math. Anal. Appl. 110 (2), pp. 327–339.

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External Links:

ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§31.15(ii).

Encodings:

BibTeX

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Zeros of Stieltjes and Van Vleck polynomials. Trans. Amer. Math. Soc. 252, pp. 197–204.

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External Links:

ISSN 0002-9947, FullText, MathReview (Pet”r Rusev), ZentralBlatt

Referenced by:

§31.15(ii).

Encodings:

BibTeX

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8: Bibliography K

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Orthogonal Polynomials on $n$-spheres: Gegenbauer, Jacobi and Heun. In Topics in Polynomials of One and Several Variables and their Applications, pp. 299–322.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§31.16(ii).

Encodings:

BibTeX

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The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.

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External Links:

Document, MathReview (M. Mikolas), ZentralBlatt

Referenced by:

§18.14(i), §18.17(ii).

Encodings:

BibTeX

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Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 30 (4), pp. 767–769.

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External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§18.21(ii).

Encodings:

BibTeX

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Askey-Wilson polynomial. Scholarpedia 7 (7), pp. 7761.

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Notes:

Electronic only

External Links:

Document, FullText

Referenced by:

§18.28(i), §18.28(i).

Encodings:

BibTeX

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Fractional integral and generalized Stieltjes transforms for hypergeometric functions as transmutation operators. SIGMA Symmetry Integrability Geom. Methods Appl. 11, pp. Paper 074, 22.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§15.14, §15.14.

Encodings:

BibTeX

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9: 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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25.2.10 $$\zeta \left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$, $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential, $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula

Proof sketch:

Derivable from (25.2.9) with $N=1$.

A&S Ref:

23.2.3

Permalink:

http://dlmf.nist.gov/25.2.E10

Encodings:

pMML, png, TeX

See also:

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

►For $B_{2k}$ see §24.2(i), and for ${\tilde{B}}_n\left(x\right)$ see §24.2(iii). …

10: 25.6 Integer Arguments

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§25.6(i) Function Values

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25.6.6 $$\zeta \left(2k+1\right)=\frac{{(-1)}^{k+1}{(2\pi )}^{2k+1}}{2(2k+1)!}{\int }_0^1{B}_{2k+1}\left(t\right)\cot \left(\pi t\right)dt,$$ $k=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $dx$: differential, $!$: factorial (as in $n!$), $\int$: integral and $k$: nonnegative integer

Source:

Nörlund (1924, (81*), p. 66); with $\nu =k$

A&S Ref:

23.2.17

Permalink:

http://dlmf.nist.gov/25.6.E6

Encodings:

pMML, png, TeX

See also:

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

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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:

pMML, png, TeX

See also:

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

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