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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)\cdots(x+n-1)

and falling factorials

(-1)^{n}{\left(-x\right)_{n}}=x(x-1)\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

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

ⓘ

Symbols:: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : 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}\mathbf{L}_{n+l}^{2l+1}(\rho_{n}).

ⓘ

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

\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,\dots

.

ⓘ

Symbols:: $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $𝐷 f$ : 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},\dots,x_{n})

gives rise to a distribution

\Lambda_{f}

defined by …

5: 37.3 Triangular Region with Weight Function $x^{\alpha}y^{\beta}(1-x-y)^{\gamma}$

… ►

37.3.5

h_{k,n}^{\alpha,\beta,\gamma}=\frac{{\left(\alpha+1\right)_{n-k}}{\left(\beta+% 1\right)_{k}}{\left(\gamma+1\right)_{k}}{\left(\beta+\gamma+2\right)_{n+k}}}{(% n-k)!\,k!\,{\left(\beta+\gamma+2\right)_{k}}{\left(\alpha+\beta+\gamma+3\right% )_{n+k}}}\*\frac{(n+k+\alpha+\beta+\gamma+2)(k+\beta+\gamma+1)}{(2n+\alpha+% \beta+\gamma+2)(2k+\beta+\gamma+1)}.

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $n$ : nonnegative integer, $h_{k,n}^{\alpha,\beta,\gamma}$ : positive constant, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : 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 and Ch.37

… ►

37.3.6

d_{j,m}=(-1)^{m-j}\*\frac{{\left(n+k+\alpha+\beta+\gamma+2\right)_{m-k}}{\left% (k+\beta+\gamma+1\right)_{j}}}{(m-k)!\,j!}\*\frac{{\left(m+k+\beta+\gamma+2% \right)_{n-m}}{\left(j+\beta+1\right)_{k-j}}}{(n-m)!\,(k-j)!}.

ⓘ

Defines:: $d_{j,m}$ : coefficient (locally)
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : 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 and Ch.37

… ►

37.3.13

\langle{U^{\alpha,\beta,\gamma}_{k,n}},{V^{\alpha,\beta,\gamma}_{j,m}}\rangle_% {\alpha,\beta,\gamma}=\frac{{\left(\alpha+1\right)_{k}}{\left(\beta+1\right)_{% n-k}}{\left(\gamma+1\right)_{n}}k!(\,n-k)!}{{\left(\alpha+\beta+\gamma+3\right% )_{2n}}}\delta_{n,m}\delta_{k,j}.

ⓘ

Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $U^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : co-monic basis element of two-variable Jacobi polynomial on the triangle, $!$ : factorial (as in $n!$ ), $V^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : monic basis element of two-variable Jacobi polynomial on the triangle, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : nonnegative integer, $n$ : nonnegative integer, $d$ : positive integer, usually $\geq 2$ , $\langle{f},{g}\rangle_{\alpha,\beta,\gamma}$ : inner product on triangular region, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : 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 $\gamma=0$ (proved)
Source:: Erdélyi et al. (1953b, 12.4(17)); only for $\gamma=0$
Permalink:: http://dlmf.nist.gov/37.3.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §37.3(iii), §37.3 and Ch.37

… ►

37.3.27

{D}_{x}P^{\alpha,\beta,\gamma}_{k,n}\left(x,y\right)=a_{k,n}^{\beta,\gamma}P^{% \alpha+1,\beta,\gamma+1}_{k-1,n-1}\left(x,y\right)+b_{k,n}^{\alpha,\beta,% \gamma}P^{\alpha+1,\beta,\gamma+1}_{k,n-1}\left(x,y\right),

ⓘ

Symbols:: $P^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{k},\NVar{n}}$ : two-variable Jacobi polynomial on the triangle, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $(\NVar{a},\NVar{b})$ : open interval, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable, $y$ : real variable, $a_{k,n}^{\beta,\gamma}$ : coefficient, $b_{k,n}^{\alpha,\beta,\gamma}$ : coefficient, $\alpha$ : real parameter ( $>-1$ ), $\beta$ : real parameter ( $>-1$ ) and $\gamma$ : real parameter ( $>-1$ )
Proved:: Xu (2017, Proposition 4.6); Use that $P^{\gamma,\beta,\alpha}_{k,n}\left(1-x-y,y\right)={\left(n+k+\alpha+\beta+% \gamma+2\right)_{n-k}}{\left(k+\alpha+\beta+1\right)_{k}}J_{k,n}^{\alpha,\beta% ,\gamma}(x,y)$ .(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 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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