binomial theorem

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1: 17.5 ${}_0{}^{}\varphi _0^{},{}_1{}^{}\varphi _0^{},{}_1{}^{}\varphi _1^{}$ Functions

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$q$-Binomial Theorem

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2: 17.2 Calculus

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§17.2(iii) Binomial Theorem

… ►In the limit as $q\to 1$, (17.2.35) reduces to the standard binomial theorem …

3: 1.2 Elementary Algebra

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Binomial Theorem

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4: 4.38 Inverse Hyperbolic Functions: Further Properties

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5: 1.9 Calculus of a Complex Variable

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DeMoivre’s Theorem

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Jordan Curve Theorem

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Cauchy’s Theorem

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Liouville’s Theorem

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Dominated Convergence Theorem

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6: 18.18 Sums

… ►See Szegő (1975, Theorems 3.1.5 and 5.7.1). … ►

§18.18(ii) Addition Theorems

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Ultraspherical

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Legendre

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§18.18(iii) Multiplication Theorems

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7: 12.13 Sums

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§12.13(i) Addition Theorems

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12.13.2 $$U(a,x+y)={\mathrm{e}}^{-\frac{1}{2}xy-\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right)y^{m}U(a+m,x),$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

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12.13.3 $$V(a,x+y)={\mathrm{e}}^{\frac{1}{2}xy+\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{a-\frac{1}{2}}{m}\right)y^{m}V(a-m,x),$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $V(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

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12.13.5 $$U(a,x\cos t+y\sin t)={\mathrm{e}}^{\frac{1}{4}{(x\sin t-y\cos t)}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right){(\tan t)}^{m}U(m+a,x)U(-m-\frac{1}{2},y),$$ $\mathrm{\Re }a\le -\frac{1}{2},0\le t\le \frac{1}{4}\pi$.

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\tan z$: tangent function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

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8: 24.4 Basic Properties

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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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Raabe’s Theorem

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9: 1.4 Calculus of One Variable

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Mean Value Theorem

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Fundamental Theorem of Calculus

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First Mean Value Theorem

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Second Mean Value Theorem

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§1.4(vi) Taylor’s Theorem for Real Variables

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10: 18.15 Asymptotic Approximations

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18.15.4_5 $${\left(\sin \frac{1}{2}\theta \right)}^{\alpha +\frac{1}{2}}{\left(\cos \frac{1}{2}\theta \right)}^{\beta +\frac{1}{2}}{P}_n^{(\alpha ,\beta )}\left(\cos \theta \right)={\pi }^{-\frac{1}{2}}{n}^{-\frac{1}{2}}\cos \left(\frac{1}{2}(2n+\alpha +\beta +1)\theta -\frac{1}{4}(2\alpha +1)\pi \right)+O\left(n^{-\frac{3}{2}}\right),$$ $\alpha ,\beta \in ℝ$,

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Symbols:

$O\left(x\right)$: order not exceeding, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\in$: element of, $ℝ$: real line, $\sin z$: sine function and $n$: nonnegative integer

Proveds:

Ismail (2009, Theorem 4.3.1); Szegő (1975, Theorem 8.21.8); with attribution to Darboux(proved)

Referenced by:

§18.15(i), Erratum (V1.2.0) §18.15

Permalink:

http://dlmf.nist.gov/18.15.E4_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.15(i), §18.15 and Ch.18

… ►The first term of this expansion also appears in Szegő (1975, Theorem 8.21.7). … ►For a bound on the error term in (18.15.10) see Szegő (1975, Theorem 8.21.11). … ►

18.15.12 $$P_n\left(\cos \theta \right)={\left(\frac{2}{\sin \theta }\right)}^{\frac{1}{2}}\sum _{m=0}^{M-1}\left(\genfrac{}{}{0.0pt}{}{-\frac{1}{2}}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m-\frac{1}{2}}{n}\right)\frac{\cos {\alpha }_{n,m}}{{(2\sin \theta )}^m}+O\left(\frac{1}{n^{M+\frac{1}{2}}}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $P_n\left(x\right)$: Legendre polynomial, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\cos z$: cosine function, $\sin z$: sine function, $m$: nonnegative integer, $n$: nonnegative integer and ${\alpha }_{n,m}$

Referenced by:

§18.15(iii)

Permalink:

http://dlmf.nist.gov/18.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(iii), §18.15 and Ch.18

… ►Another expansion follows from (18.15.10) by taking $\lambda =\frac{1}{2}$; see Szegő (1975, Theorem 8.21.5). …