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

… ►

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 …

5: 37.3 Triangular Region with Weight Function

… ►

37.3.5

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $k$: nonnegative integer, $n$: nonnegative integer, : positive constant, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proved:

Dunkl and Xu (2014, Proposition 2.4.1)(proved)

Referenced by:

§37.3(ii)

Permalink:

http://dlmf.nist.gov/37.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §37.3(ii), §37.3, Part 37.II and Ch.37

… ►

37.3.6

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Defines:

$d_{j,m}$: coefficient (locally)

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proof sketch:

Apply the symmetry in the second row of Table 18.6.1 to the second Jacobi polynomial in (37.3.3). Then apply (18.5.7) twice to the resulting right-hand side of (37.3.3).

Permalink:

http://dlmf.nist.gov/37.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §37.3(ii), §37.3, Part 37.II and Ch.37

… ►

37.3.13

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Symbols:

${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , : co-monic basis element of two-variable Jacobi polynomial on the triangular region, : inner product on triangular region, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proveds:

Engelis (1964); Fackerell and Littler (1974, (11)); Dunkl and Xu (2014, Proposition 5.3.3); for $d=2$; Appell and Kampé de Fériet (1926, §XXXVI); only for (proved)

Source:

Erdélyi et al. (1953b, 12.4(17)); only for

Permalink:

http://dlmf.nist.gov/37.3.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.3(iii), §37.3, Part 37.II and Ch.37

… ►

37.3.27

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $(a,b)$: open interval, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $y$: real variable, : coefficient, : coefficient, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proved:

Xu (2017, Proposition 4.6); Use that .(proved)

Referenced by:

(37.3.28)

Permalink:

http://dlmf.nist.gov/37.3.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §37.3(v), §37.3, Part 37.II and Ch.37

… ►The three-term relations (37.2.7) give rise to a three-term and a nine-term relation as follows: …

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