partitional shifted factorial

(0.003 seconds)

10 matching pages

1: 35.4 Partitions and Zonal Polynomials
Also, $|\kappa|$ denotes $k_{1}+\dots+k_{m}$, the weight of $\kappa$; $\ell(\kappa)$ denotes the number of nonzero $k_{j}$; $a+\kappa$ denotes the vector $(a+k_{1},\dots,a+k_{m})$. The partitional shifted factorial is given by
35.4.2 $Z_{\kappa}\left(\mathbf{I}\right)=|\kappa|!\,2^{2|\kappa|}\,{\left[m/2\right]_% {\kappa}}\frac{\prod\limits_{1\leq j
35.4.9 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left|\mathbf{X}\right|^{a% -\frac{1}{2}(m+1)}\*\left|\mathbf{I}-\mathbf{X}\right|^{b-\frac{1}{2}(m+1)}Z_{% \kappa}\left(\mathbf{T}\mathbf{X}\right)\mathrm{d}{\mathbf{X}}=\frac{{\left[a% \right]_{\kappa}}}{{\left[a+b\right]_{\kappa}}}\mathrm{B}_{m}\left(a,b\right)Z% _{\kappa}\left(\mathbf{T}\right).$
2: 35.1 Special Notation
 $a,b$ complex variables. … partitional shifted factorial (§35.4(i)). …
3: 35.5 Bessel Functions of Matrix Argument
35.5.2 $A_{\nu}\left(\mathbf{T}\right)=A_{\nu}\left(\boldsymbol{{0}}\right)\sum_{k=0}^% {\infty}\frac{(-1)^{k}}{k!}\sum_{|\kappa|=k}\frac{1}{{\left[\nu+\frac{1}{2}(m+% 1)\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
4: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.1 ${{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}{k!% }\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}}{{\left[b\right]_{\kappa}}}Z% _{\kappa}\left(\mathbf{T}\right).$
5: 35.8 Generalized Hypergeometric Functions of Matrix Argument
35.8.1 ${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{T}\right% )=\sum_{k=0}^{\infty}\frac{1}{k!}\sum_{|\kappa|=k}\frac{{\left[a_{1}\right]_{% \kappa}}\cdots{\left[a_{p}\right]_{\kappa}}}{{\left[b_{1}\right]_{\kappa}}% \cdots{\left[b_{q}\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right).$
6: 35.7 Gaussian Hypergeometric Function of Matrix Argument
35.7.1 ${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}% {k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}% {{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},$ $-c+\frac{1}{2}(j+1)\notin\mathbb{N}$, $1\leq j\leq m$; $\|\mathbf{T}\|<1$.
8: 26.17 The Twelvefold Way
In this table ${\left(k\right)_{n}}$ is Pochhammer’s symbol, and $S\left(n,k\right)$ and $p_{k}\left(n\right)$ are defined in §§26.8(i) and 26.9(i). …
9: 27.14 Unrestricted Partitions
§27.14(iii) Asymptotic Formulas
For example, $p\left(10\right)=42,p\left(100\right)$ = $1905\;69292$, and $p\left(200\right)=397\;29990\;29388$. …
10: 26.8 Set Partitions: Stirling Numbers
§26.8 Set Partitions: Stirling Numbers
$S\left(n,k\right)$ denotes the Stirling number of the second kind: the number of partitions of $\{1,2,\ldots,n\}$ into exactly $k$ nonempty subsets. …
26.8.7 $\sum_{k=0}^{n}s\left(n,k\right)x^{k}={\left(x-n+1\right)_{n}},$