partitional shifted factorial

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1: 35.4 Partitions and Zonal Polynomials

… ►Also, $|\kappa |$ denotes $k_1+\mathrm{\cdots }+k_m$, the weight of $\kappa$; $\mathrm{\ell }(\kappa )$ denotes the number of nonzero $k_j$; $a+\kappa$ denotes the vector $(a+k_1,\mathrm{\dots },a+k_m)$. ►The partitional shifted factorial is given by ►

35.4.1 $${\left[a\right]}_{\kappa }=\frac{{\mathrm{\Gamma }}_m\left(a+\kappa \right)}{{\mathrm{\Gamma }}_m\left(a\right)}=\prod _{j=1}^m{\left(a-\frac{1}{2}(j-1)\right)}_{k_j},$$

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Defines:

${\left[a\right]}_{\kappa }$: partitional shifted factorial

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

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35.4.2

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Symbols:

$!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer, $\kappa$: vector of nonnegative integers and $\mathrm{\ell }(\kappa )$: number of nonzeros

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

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35.4.9

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Symbols:

$\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer, $𝟎$: $m\times m$ matrix of zeros and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

2: 35.1 Special Notation

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$a,b$	complex variables.
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${\left[a\right]}_{\kappa }$	partitional shifted factorial (§35.4(i)).
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3: 35.5 Bessel Functions of Matrix Argument

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35.5.2 $$A_{\nu }\left(𝐓\right)=A_{\nu }\left(𝟎\right)\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\sum _{|\kappa |=k}\frac{1}{{\left[\nu +\frac{1}{2}(m+1)\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right),$$ $\nu \in ℂ$, $𝐓\in 𝓢$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, $!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix, $k$: nonnegative integer, $m$: positive integer, $𝟎$: $m\times m$ matrix of zeros and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(i), §35.5 and Ch.35

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4: 35.6 Confluent Hypergeometric Functions of Matrix Argument

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35.6.1 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{b};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }}{{\left[b\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $k$: nonnegative integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(i), §35.6 and Ch.35

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5: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.1 $${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a_1\right]}_{\kappa }\mathrm{\cdots }{\left[a_p\right]}_{\kappa }}{{\left[b_1\right]}_{\kappa }\mathrm{\cdots }{\left[b_q\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right).$$

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Defines:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument

Symbols:

$!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $k$: nonnegative integer, $p$: nonnegative integer, $q$: nonnegative integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.10, §35.10, §35.8(i), §35.8(i), §35.8(i), §35.8(i)

Permalink:

http://dlmf.nist.gov/35.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(i), §35.8 and Ch.35

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6: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.1 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }{\left[b\right]}_{\kappa }}{{\left[c\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right),$$ $-c+\frac{1}{2}(j+1)\notin \mathbb{N}$, $1\le j\le m$; .

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $!$: factorial (as in $n!$), $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $\Vert 𝐓\Vert$: spectral norm of $𝐓$ , $c$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(i), §35.7 and Ch.35

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7: 26.12 Plane Partitions

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26.12.17 $$\prod _{h=0}^{r-1}\frac{(3h+1)!}{(r+h)!}.$$

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Symbols:

$!$: factorial (as in $n!$), $r$: positive integer and $h$: positive integer

Permalink:

http://dlmf.nist.gov/26.12.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §26.12(i), §26.12 and Ch.26

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8: 26.17 The Twelvefold Way

… ►In this table ${\left(k\right)}_n$ is Pochhammer’s symbol, and $S(n,k)$ and $p_k\left(n\right)$ are defined in §§26.8(i) and 26.9(i). … ►

Table 26.17.1: The twelvefold way.

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elements of $N$	elements of $K$	$f$ unrestricted	$f$ one-to-one	$f$ onto
labeled	labeled	$k^n$	${\left(k-n+1\right)}_n$	$k!S(n,k)$
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unlabeled	unlabeled	$p_k\left(n\right)$		$p_k\left(n\right)-p_{k-1}\left(n\right)$

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9: 27.14 Unrestricted Partitions

§27.14 Unrestricted Partitions

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§27.14(i) Partition Functions

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§27.14(iii) Asymptotic Formulas

… ►For example, $p\left(10\right)=42,p\left(100\right)$ = $\mathrm{1905\hspace{0.33em}69292}$, and $p\left(200\right)=\mathrm{397\hspace{0.33em}29990\hspace{0.33em}29388}$. … ►

§27.14(v) Divisibility Properties

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10: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

… ► $S(n,k)$ denotes the Stirling number of the second kind: the number of partitions of $\{1,2,\mathrm{\dots },n\}$ into exactly $k$ nonempty subsets. … ►

26.8.7 $$\sum _{k=0}^{n}s(n,k)x^k={\left(x-n+1\right)}_n,$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $s(n,k)$: Stirling number of the first kind, $x$: real variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.3 (in slightly different form)

Permalink:

http://dlmf.nist.gov/26.8.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §26.8(ii), §26.8 and Ch.26

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26.8.10 $$\sum _{k=1}^{n}S(n,k){\left(x-k+1\right)}_k=x^n,$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $S(n,k)$: Stirling number of the second kind, $x$: real variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.4

Referenced by:

§26.15

Permalink:

http://dlmf.nist.gov/26.8.E10

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §26.8(ii), §26.8 and Ch.26

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