binomial expansion

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1: 4.6 Power Series

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Binomial Expansion

… ►Note that (4.6.7) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6).

2: 28.8 Asymptotic Expansions for Large $q$

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28.8.4 $$U_m(\xi )\sim D_m\left(\xi \right)-\frac{1}{2^{6}h}\left(D_{m+4}\left(\xi \right)-4!\left(\genfrac{}{}{0pt}{}{m}{4}\right)D_{m-4}\left(\xi \right)\right)+\frac{1}{2^{13}{h}^2}\left(D_{m+8}\left(\xi \right)-2^5(m+2)D_{m+4}\left(\xi \right)+4!{\mathrm{\hspace{0.17em}2}}^5(m-1)\left(\genfrac{}{}{0pt}{}{m}{4}\right)D_{m-4}\left(\xi \right)+8!\left(\genfrac{}{}{0.0pt}{}{m}{8}\right)D_{m-8}\left(\xi \right)\right)+\mathrm{\cdots },$$

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Defines:

$U_m(\xi )$: function (locally)

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable

A&S Ref:

20.9.17 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

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

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Defines:

$V_m(\xi )$: function (locally)

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable

A&S Ref:

20.9.18 (in slightly different form)

Referenced by:

Erratum (V1.0.16) for Equation (28.8.5)

Permalink:

http://dlmf.nist.gov/28.8.E5

Encodings:

pMML, png, TeX

Error (effective with 1.0.16):

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

Suggested 2017-02-02 by Daniel Karlsson

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

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3: 8.17 Incomplete Beta Functions

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8.17.5 $$I_x(m,n-m+1)=\sum _{j=m}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)x^j{(1-x)}^{n-j},$$ $m,n$ positive integers; .

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Symbols:

$I_x(a,b)$: incomplete beta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $n$: nonnegative integer and $x$: variable

A&S Ref:

6.6.4 (Relation to the BinomialExpansion)

Referenced by:

§8.17(i), §8.17(vii), Erratum (V1.0.5) for Subsection 8.17(i)

Permalink:

http://dlmf.nist.gov/8.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(i), §8.17 and Ch.8

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4: 2.6 Distributional Methods

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2.6.6 $$S(x)\sim \frac{2\pi }{\sqrt{3}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\left(\genfrac{}{}{0.0pt}{}{-\frac{1}{3}}{s}\right)x^{-s-(1/3)},$$ $x\to \mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter and $S(x)$: integral

Referenced by:

§2.6(i)

Permalink:

http://dlmf.nist.gov/2.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(i), §2.6 and Ch.2

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2.6.7 $$S(x)\sim \frac{2\pi }{\sqrt{3}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\left(\genfrac{}{}{0.0pt}{}{-\frac{1}{3}}{s}\right)x^{-s-(1/3)}-\sum _{s=1}^{\mathrm{\infty }}\frac{3^s(s-1)!}{2\cdot 5\mathrm{\cdots }(3s-1)}{x}^{-s};$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $S(x)$: integral

Referenced by:

§2.6(i), §2.6(ii)

Permalink:

http://dlmf.nist.gov/2.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(i), §2.6 and Ch.2

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5: 2.10 Sums and Sequences

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2.10.7 $$\sum _{j=1}^{n-1}{j}^{\alpha }\sim \zeta \left(-\alpha \right)+\frac{n^{\alpha +1}}{\alpha +1}\sum _{s=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{\alpha +1}{s}\right)\frac{B_s}{n^s},$$ $n\to \mathrm{\infty }$,

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $n$: integer

Referenced by:

§2.10(i), §2.10(ii)

Permalink:

http://dlmf.nist.gov/2.10.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.10(i), §2.10(i), §2.10 and Ch.2

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6: 1.10 Functions of a Complex Variable

… ►Note that (1.10.4) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6). …

7: 24.4 Basic Properties

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§24.4(iv) Finite Expansions

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24.4.12 $$B_n\left(x+h\right)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k\left(x\right)h^{n-k},$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.7

Permalink:

http://dlmf.nist.gov/24.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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24.4.13 $$E_n\left(x+h\right)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(x\right)h^{n-k},$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.7

Permalink:

http://dlmf.nist.gov/24.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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24.4.14 $$E_{n-1}\left(x\right)=\frac{2}{n}\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)(1-2^k)B_k{x}^{n-k},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.4.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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24.4.15 $$B_{2n}=\frac{2n}{2^{2n}(2^{2n}-1)}\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{2n-1}{2k}\right)E_{2k},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.4.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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8: 18.18 Sums

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§18.18(i) Series Expansions of Arbitrary Functions

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Legendre

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Laguerre

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Hermite

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Ultraspherical

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9: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(i) Numerical Use of Asymptotic Expansions

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§2.11(iii) Exponentially-Improved Expansions

… ►In this way we arrive at hyperasymptotic expansions. … ► …

10: 13.8 Asymptotic Approximations for Large Parameters

… ►For the parabolic cylinder function $U$ see §12.2, and for an extension to an asymptotic expansion see Temme (1978). … ►For other asymptotic expansions for large $b$ and $z$ see López and Pagola (2010). ►For more asymptotic expansions for the cases $b\to \pm \mathrm{\infty }$ see Temme (2015, §§10.4 and 22.5) … ►For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i). … ►These results follow from Temme (2022), which can also be used to obtain more terms in the expansions. …