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11: 8.11 Asymptotic Approximations and Expansions

… ►This reference also contains explicit formulas for $b_k(\lambda )$ in terms of Stirling numbers and for the case $\lambda >1$ an asymptotic expansion for $b_k(\lambda )$ as $k\to \mathrm{\infty }$. … ►This reference also contains explicit formulas for the coefficients in terms of Stirling numbers. …

12: 18.13 Continued Fractions

… ►The following formulae are explicit cases of (18.2.34)–(18.2.36); this area is fully explored in §§18.30(vi) and 18.30(vii). …

13: Bibliography W

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Some explicit Padé approximants for the function ${\mathrm{\Phi }}^{\prime }/\mathrm{\Phi }$ and a related quadrature formula involving Bessel functions. SIAM J. Math. Anal. 16 (4), pp. 887–895.

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External Links:

ISSN 0036-1410, Document, MathReview (Wyman Fair), ZentralBlatt

Referenced by:

§13.31(ii).

Encodings:

BibTeX

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Explicit formulas for the associated Jacobi polynomials and some applications. Canad. J. Math. 39 (4), pp. 983–1000.

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External Links:

ISSN 0008-414X, MathReview (W. A. Al-Salam), ZentralBlatt

Referenced by:

§18.30(i), §18.30.

Encodings:

BibTeX

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14: Bibliography G

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Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.

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External Links:

FullText, MathReview (B. M. Stewart), ZentralBlatt

Referenced by:

§24.15(iii), §24.6.

Encodings:

BibTeX

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15: 5.11 Asymptotic Expansions

… ►For explicit formulas for $g_k$ in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of $g_k$ as $k\to \mathrm{\infty }$ see Boyd (1994) and Nemes (2015a). …

16: 18.38 Mathematical Applications

… ►See Koornwinder (2007a, (3.13), (4.9), (4.10)) for explicit formulas. …

17: 4.13 Lambert $W$-Function

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4.13.4_1 $$\frac{d^{n}W}{{dz}^n}=\frac{{\mathrm{e}}^{-nW}{p}_{n-1}(W)}{{\left(1+W\right)}^{2n-1}},$$ $n=1,2,3,\mathrm{\dots }$,

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $n$: integer and $z$: complex variable

Notes:

See Corless et al. (1997, formula (7)).

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

… ►Explicit representations for the $p_n(x)$ are given in Kalugin and Jeffrey (2011). … ►

4.13.5_1 $${\left(\frac{W_0\left(z\right)}{z}\right)}^a={\mathrm{e}}^{-a{W}_0\left(z\right)}=\sum _{n=0}^{\mathrm{\infty }}\frac{a{\left(n+a\right)}^{n-1}}{n!}{\left(-z\right)}^n,$$ , $a\in ℂ$.

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $ℂ$: complex plane, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $a$: real or complex constant and $z$: complex variable

Notes:

See Corless et al. (1996, formula (2.36)).

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E5_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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4.13.5_2 $$\frac{1}{1+W_0\left(-z\right)}=\sum _{n=0}^{\mathrm{\infty }}\frac{n^n}{n!}{z}^n,$$ .

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer and $z$: complex variable

Notes:

See Corless et al. (1997, formula (14)).

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E5_2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.13 and Ch.4

… ►See Jeffrey and Murdoch (2017) for an explicit representation for the $c_n$ in terms of associated Stirling numbers. …

18: Errata

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§4.13 has been enlarged. The Lambert $W$-function is multi-valued and we use the notation $W_k\left(x\right)$, $k\in \mathbb{Z}$, for the branches. The original two solutions are identified via $\mathrm{Wp}\left(x\right)=W_0\left(x\right)$ and $\mathrm{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$.

Other changes are the introduction of the Wright $\omega$-function and tree $T$-function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for $\frac{d^{n}W}{{dz}^n}$, additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at $z=-{\mathrm{e}}^{-1}$ in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert $W$-functions in the end of the section.

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19: 18.20 Hahn Class: Explicit Representations

§18.20 Hahn Class: Explicit Representations

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§18.20(i) Rodrigues Formulas

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Table 18.20.1: Krawtchouk, Meixner, and Charlier OP’s: Rodrigues formulas (18.20.1).

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$p_n(x)$	$F(x)$	${\kappa }_n$
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20: 18.35 Pollaczek Polynomials

… ►Thus type 3 with $c=0$ reduces to type 2, and type 3 with $c=0$ and $\lambda =\frac{1}{2}$ reduces to type 1, also in subsequent formulas. … ►we have the explicit representations ►

18.35.4 $$P_n^{(\lambda )}(\cos \theta ;a,b)=\frac{{\left(\lambda -\mathrm{i}{\tau }_{a,b}(\theta )\right)}_n}{n!}{\mathrm{e}}^{\mathrm{i}n\theta }F(\genfrac{}{}{0pt}{}{-n,\lambda +\mathrm{i}{\tau }_{a,b}(\theta )}{-n-\lambda +1+\mathrm{i}{\tau }_{a,b}(\theta )};{\mathrm{e}}^{-2\mathrm{i}\theta })=\sum _{\mathrm{\ell }=0}^n\frac{{\left(\lambda +\mathrm{i}{\tau }_{a,b}(\theta )\right)}_{\mathrm{\ell }}}{\mathrm{\ell }!}\frac{{\left(\lambda -\mathrm{i}{\tau }_{a,b}(\theta )\right)}_{n-\mathrm{\ell }}}{(n-\mathrm{\ell })!}{\mathrm{e}}^{\mathrm{i}(n-2\mathrm{\ell })\theta },$$

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Defines:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and ${\tau }_{a,b}(\theta )$

Proved:

Ismail (2009, (5.4.10)); In the cited formula, in the lower parameter of the hypergeometric function, $-\lambda$ should be replaced by $-\lambda +1$.(proved)

Referenced by:

(18.35.4_5)

Permalink:

http://dlmf.nist.gov/18.35.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(i), §18.35 and Ch.18

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