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: 37.11 Spherical Harmonics

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37.11.5 $$N_n^d=dim{\mathscr{H}}_n^{0,d}=dim{\mathscr{H}}_n^d=\frac{2n+d-2}{n+d-2}\left(\genfrac{}{}{0.0pt}{}{n+d-2}{n}\right),$$ $n\in {\mathbb{N}}_0$, $d\ge 2$.

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\in$: element of, $\mathbb{N}$: set of all positive integers, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , ${\mathbb{N}}_0$: set of all nonnegative integers, ${\mathscr{H}}_n^{0,d}$: space of spherical harmonics of degree $n$ on ${𝕊}^{d-1}$ and ${\mathscr{H}}_n^d$: complex linear space

Proved:

Dunkl and Xu (2014, Theorem 4.1.2)(proved)

Referenced by:

§37.12(i)

Permalink:

http://dlmf.nist.gov/37.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.11, Part 37.III and Ch.37

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6: null

error generating summary

7: 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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8: 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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9: 37.14 Orthogonal Polynomials on the Simplex

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37.14.8 $$V_{𝝂}^{𝜶}(𝐱)=\sum _{𝝁\le 𝝂}{(-1)}^{|𝝂|+|𝝁|}\left(\genfrac{}{}{0.0pt}{}{𝝂}{𝝁}\right)\frac{{\left(𝜶+\mathrm{𝟏}\right)}_{𝝂}{\left(|𝜶|+d\right)}_{|𝝂|+|𝝁|}}{{\left(𝜶+\mathrm{𝟏}\right)}_{𝝁}{\left(|𝜶|+d\right)}_{2|𝝂|}}{𝐱}^{𝝁},,$$ ${\left(𝜶+\mathrm{𝟏}\right)}_{𝝂}={\prod }_{\mathrm{\ell }=1}^d{\left({\alpha }_{\mathrm{\ell }}+1\right)}_{{\nu }_{\mathrm{\ell }}}$ $𝝂\in {\mathbb{N}}_0^d$, $|𝝂|=n$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\in$: element of, $\mathbb{N}$: set of all positive integers, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $\mathrm{\ell }$: positive integer, $𝐱$: vector in $d$-dimensional Euclidean space and ${\mathbb{N}}_0$: set of all nonnegative integers

Source:

Dunkl and Xu (2014, Theorem 5.3.4); There replace ${\kappa }_{\mathrm{\ell }}$ by ${\alpha }_{\mathrm{\ell }}+\frac{1}{2}$ ($\mathrm{\ell }=1,\mathrm{\dots },d+1$).

Referenced by:

§37.15(vii)

Permalink:

http://dlmf.nist.gov/37.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §37.14(iii), §37.14, Part 37.III and Ch.37

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10: 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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