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11: 5.11 Asymptotic Expansions

§5.11 Asymptotic Expansions

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§5.11(i) Poincaré-Type Expansions

… ►and … ►

§5.11(iii) Ratios

… ►

12: 13.14 Definitions and Basic Properties

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13.14.9 $$W_{\kappa ,\pm \frac{1}{2}n}\left(z\right)={(-1)}^{\kappa -\frac{1}{2}n-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}z}{z}^{\frac{1}{2}n+\frac{1}{2}}\sum _{k=0}^{\kappa -\frac{1}{2}n-\frac{1}{2}}\left(\genfrac{}{}{0.0pt}{}{\kappa -\frac{1}{2}n-\frac{1}{2}}{k}\right){\left(n+1+k\right)}_{\kappa -k-\frac{1}{2}n-\frac{1}{2}}{(-z)}^k,$$ $\kappa -\frac{1}{2}n-\frac{1}{2}=0,1,2,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.14(i)

Permalink:

http://dlmf.nist.gov/13.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(i), §13.14(i), §13.14 and Ch.13

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13: 2.9 Difference Equations

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$f(n)\sim {\displaystyle \sum _{s=0}^{\mathrm{\infty }}}{\displaystyle \frac{f_s}{n^s}},$

… ►

2.9.7 $${\rho }_j(f_0+2{\rho }_j)s{a}_{s,j}=\sum _{r=1}^s\left({\rho }_j^2{2}^{r+1}\left(\genfrac{}{}{0.0pt}{}{{\alpha }_j+r-s}{r+1}\right)+{\rho }_j\sum _{q=0}^{r+1}\left(\genfrac{}{}{0.0pt}{}{{\alpha }_j+r-s}{r+1-q}\right)f_q+g_{r+1}\right)a_{s-r,j},$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $f_s$: coefficients, $g_s$: coefficients, $a_{s,j}$: coefficients and ${\rho }_j$: roots

Referenced by:

§2.9(ii)

Permalink:

http://dlmf.nist.gov/2.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(i), §2.9 and Ch.2

… ►For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). … ►

2.9.12 $$w_j(n)\sim {\rho }^n{n}^{{\alpha }_j}\sum _{s=0}^{\mathrm{\infty }}\frac{a_{s,j}}{n^s},$$ $n\to \mathrm{\infty }$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\rho$: roots and $w_j(n)$: solutions

Referenced by:

§2.9(ii)

Permalink:

http://dlmf.nist.gov/2.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §2.9(ii), §2.9 and Ch.2

… ►For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). …

14: Errata

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28.8.5 $$V_m(\xi )\sim \frac{1}{2^{4}h}\left(-D_{m+2}\left(\xi \right)-m(m-1)D_{m-2}\left(\xi \right)\right)+\frac{1}{2^{10}{h}^2}\left(D_{m+6}\left(\xi \right)+(m^2-25m-36)D_{m+2}\left(\xi \right)-m(m-1)(m^2+27m-10)D_{m-2}\left(\xi \right)-6!\left(\genfrac{}{}{0.0pt}{}{m}{6}\right)D_{m-6}\left(\xi \right)\right)+\mathrm{\cdots }$$

Originally the $-$ in front of the $6!$ was given incorrectly as $+$.

Reported 2017-02-02 by Daniel Karlsson.

…

15: 18.15 Asymptotic Approximations

… ►For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term in a related expansion see Wong and Zhao (2003). …These expansions are in terms of Whittaker functions (§13.14). … ►The first term of this expansion also appears in Szegő (1975, Theorem 8.21.7). … ►These expansions are in terms of Bessel functions and modified Bessel functions, respectively. … ►For an error bound for the first term in the Airy-function expansions see Olver (1997b, p. 403). …

16: 25.2 Definition and Expansions

§25.2 Definition and Expansions

… ►

§25.2(ii) Other Infinite Series

… ►For further expansions of functions similar to (25.2.1) (Dirichlet series) see §27.4. … ►

25.2.9 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-\frac{1}{2}{N}^{-s}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}{N}^{1-s-2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_N^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$; $n,N=1,2,3,\mathrm{\dots }$.

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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 of $x$ , $\int$: 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, improper integral

Proof sketch:

Derivable from (25.2.8) by repeated integration by parts.

Referenced by:

§25.18(i), (25.2.10)

Permalink:

http://dlmf.nist.gov/25.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §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 of $x$ , $\int$: 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

…

17: 25.11 Hurwitz Zeta Function

… ►The function $\zeta (s,a)$ was introduced in Hurwitz (1882) and defined by the series expansion … ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►

§25.11(xii) $a$-Asymptotic Behavior

… ►As $a\to \mathrm{\infty }$ in the sector , with $s(\ne 1)$ and $\delta$ fixed, we have the asymptotic expansion … ►Similarly, as $a\to \mathrm{\infty }$ in the sector , …

18: 3.9 Acceleration of Convergence

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3.9.4 $${\mathrm{\Delta }}^k{a}_0=\sum _{m=0}^k{(-1)}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)a_{k-m}.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $\mathrm{\Delta }$: forward difference operator

A&S Ref:

3.6.27

Permalink:

http://dlmf.nist.gov/3.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(ii), §3.9 and Ch.3

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3.9.13 $${ℒ}_k^{(n)}(s)=\frac{{\sum }_{j=0}^k{(-1)}^j\left(\genfrac{}{}{0.0pt}{}{k}{j}\right)c_{j,k,n}{s}_{n+j}/a_{n+j+1}}{{\sum }_{j=0}^k{(-1)}^j\left(\genfrac{}{}{0.0pt}{}{k}{j}\right)c_{j,k,n}/a_{n+j+1}},$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $s_n$: sequence and $c_{j,k,n}$: coefficient

Referenced by:

§3.9(v)

Permalink:

http://dlmf.nist.gov/3.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §3.9(v), §3.9 and Ch.3

… ►For applications to asymptotic expansions, see §2.11(vi), Olver (1997b, pp. 540–543), and Weniger (1989, 2003).

19: Bibliography K

… ►

Hypergeometric expansions of Heun polynomials. SIAM J. Math. Anal. 22 (5), pp. 1450–1459.

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External Links:

ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt

Referenced by:

§31.16(ii).

Encodings:

BibTeX

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Addendum: “Hypergeometric expansions of Heun polynomials”. SIAM J. Math. Anal. 22 (6), pp. 1803.

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External Links:

ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt

Referenced by:

§31.16(ii).

Encodings:

BibTeX

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Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.

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External Links:

ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§19.12, §19.5.

Encodings:

BibTeX

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Asymptotic expansions of certain $q$-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

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External Links:

ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§17.2(i).

Encodings:

BibTeX

… ►

HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

…

20: 31.15 Stieltjes Polynomials

… ►There exist at most $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ polynomials $V(z)$ of degree not exceeding $N-2$ such that for $\mathrm{\Phi }(z)=V(z)$, (31.15.1) has a polynomial solution $w=S(z)$ of degree $n$. … ►then there are exactly $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ polynomials $S(z)$, each of which corresponds to each of the $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ ways of distributing its $n$ zeros among $N-1$ intervals $(a_j,a_{j+1})$, $j=1,2,\mathrm{\dots },N-1$. … ►For further details and for the expansions of analytic functions in this basis see Volkmer (1999).