binomial expansion

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1: 4.6 Power Series
BinomialExpansion
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 $q$
28.8.4 $U_{m}(\xi)\sim D_{m}\left(\xi\right)-\frac{1}{2^{6}h}\left(D_{m+4}\left(\xi% \right)-4!\dbinom{m}{4}D_{m-4}\left(\xi\right)\right)+\frac{1}{2^{13}h^{2}}% \left(D_{m+8}\left(\xi\right)-2^{5}(m+2)D_{m+4}\left(\xi\right)+4!\,2^{5}(m-1)% \dbinom{m}{4}D_{m-4}\left(\xi\right)+8!\genfrac{(}{)}{0.0pt}{}{m}{8}D_{m-8}% \left(\xi\right)\right)+\cdots,$
28.8.5 $V_{m}(\xi)\sim\frac{1}{2^{4}h}\bigg{(}-D_{m+2}\left(\xi\right)-m(m-1)D_{m-2}% \left(\xi\right)\bigg{)}+\frac{1}{2^{10}h^{2}}\left(D_{m+6}\left(\xi\right)+(m% ^{2}-25m-36)D_{m+2}\left(\xi\right)-m(m-1)(m^{2}+27m-10)D_{m-2}\left(\xi\right% )-6!\genfrac{(}{)}{0.0pt}{}{m}{6}D_{m-6}\left(\xi\right)\right)+\cdots,$
3: 8.17 Incomplete Beta Functions
8.17.5 $I_{x}\left(m,n-m+1\right)=\sum_{j=m}^{n}\genfrac{(}{)}{0.0pt}{}{n}{j}x^{j}(1-x% )^{n-j},$ $m,n$ positive integers; $0\leq x<1$.
4: 2.6 Distributional Methods
2.6.6 $S(x)\sim\frac{2\pi}{\sqrt{3}}\sum_{s=0}^{\infty}(-1)^{s}{\genfrac{(}{)}{0.0pt}% {}{-\frac{1}{3}}{s}}x^{-s-(1/3)},$ $x\to\infty$.
2.6.7 $S(x)\sim\frac{2\pi}{\sqrt{3}}\sum_{s=0}^{\infty}(-1)^{s}{\genfrac{(}{)}{0.0pt}% {}{-\frac{1}{3}}{s}}x^{-s-(1/3)}-\sum_{s=1}^{\infty}\frac{3^{s}(s-1)!}{2\cdot 5% \cdots(3s-1)}x^{-s};$
5: 2.10 Sums and Sequences
2.10.7 $\sum_{j=1}^{n-1}j^{\alpha}\sim\zeta\left(-\alpha\right)+\frac{n^{\alpha+1}}{% \alpha+1}\sum_{s=0}^{\infty}\genfrac{(}{)}{0.0pt}{}{\alpha+1}{s}\frac{B_{s}}{n% ^{s}},$ $n\to\infty$,
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
§24.4(iv) Finite Expansions
24.4.14 $E_{n-1}\left(x\right)=\frac{2}{n}\sum_{k=0}^{n}{n\choose k}(1-2^{k})B_{k}x^{n-% k},$
24.4.15 $B_{2n}=\frac{2n}{2^{2n}(2^{2n}-1)}\sum_{k=0}^{n-1}{2n-1\choose 2k}E_{2k},$
9: 2.11 Remainder Terms; Stokes Phenomenon
§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 $b\to\pm\infty$ 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). …
$p_{k}(z)=\sum_{s=0}^{k}\genfrac{(}{)}{0.0pt}{}{k}{s}{\left(1-b+s\right)_{k-s}}% z^{s}c_{k+s}(z),$