factorials (rising or falling)

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1: 26.1 Special Notation

… ►Many combinatorics references use the rising and falling factorials: …

2: 5.2 Definitions

… ►Pochhammer symbols (rising factorials) ${\left(x\right)}_n=x(x+1)\mathrm{\cdots }(x+n-1)$ and falling factorials ${(-1)}^n{\left(-x\right)}_n=x(x-1)\mathrm{\cdots }(x-n+1)$ can be expressed in terms of each other via …

3: 18.39 Applications in the Physical Sciences

… ►Here the $H_n\left(x\right)$ are Hermite polynomials, $w(x)={\mathrm{e}}^{-x^2}$, and $h_n=2^{n}n!\sqrt{\pi }$. … ►see Bethe and Salpeter (1957, p. 13), Pauling and Wilson (1985, pp. 130, 131); and noting that this differs from the Rodrigues formula of (18.5.5) for the Laguerre OP’s, in the omission of an $n!$ in the denominator. … ►

18.39.40 $${𝐋}_{p+m}^m(\rho )={(-1)}^m(p+m)!L_p^{(m)}\left(\rho \right),$$

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$) and $m$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.39.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

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18.39.42 $$R_{n,l}(r)=\frac{2}{n^2}\sqrt{\frac{Z^3(n-l-1)!}{{\left((n+l)!\right)}^3}}{\mathrm{e}}^{-{\rho }_n/2}{\rho }_n^l{𝐋}_{n+l}^{2l+1}({\rho }_n).$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $l$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§18.39(ii), §18.39(ii)

Permalink:

http://dlmf.nist.gov/18.39.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

… ►For applications of Legendre polynomials in fluid dynamics to study the flow around the outside of a puff of hot gas rising through the air, see Paterson (1983). …

4: 1.16 Distributions

… ►We denote a regular distribution by ${\mathrm{\Lambda }}_f$, or simply $f$, where $f$ is the function giving rise to the distribution. … ►

1.16.23 $${(-1)}^{n}n!x_{+}^{-1-n}={𝐷}^{(n+1)}{\ln }_{+}x,$$ $n=0,1,2,\mathrm{\dots }$.

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Symbols:

$!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $𝐷f$: distributional derivative

Permalink:

http://dlmf.nist.gov/1.16.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(iv), §1.16 and Ch.1

… ►A locally integrable function $f(x)=f(x_1,x_2,\mathrm{\dots },x_n)$ gives rise to a distribution ${\mathrm{\Lambda }}_f$ defined by …