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1: 26.12 Plane Partitions

… ►An equivalent definition is that a plane partition is a finite subset of

\mathbb{N}\times\mathbb{N}\times\mathbb{N}

with the property that if

(r,s,t)\in\pi

and

(1,1,1)\leq(h,j,k)\leq(r,s,t)

, then

(h,j,k)

must be an element of

\pi

. … ►

26.12.20

\sum_{\pi\subseteq\mathbb{N}\times\mathbb{N}\times\mathbb{N}}q^{|\pi|}=\prod_{% k=1}^{\infty}\frac{1}{(1-q^{k})^{k}},

ⓘ

Symbols:: $\mathbb{N}$ : set of all positive integers, $\subseteq$ : is, or is contained in, $k$ : nonnegative integer, $\pi$ : plane partition and $q$ : parameter
Permalink:: http://dlmf.nist.gov/26.12.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §26.12(ii), §26.12 and Ch.26

…

2: 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

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $!$ : factorial (as in $n!$ ), $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $\|\mathbf{T}\|$ : spectral norm of $\mathbf{T}$ , $c$ : 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

… ►

35.7.8

{{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(c-a-b\right)}{\Gamma_{m}\left(c-a\right)\Gamma_{m}\left% (c-b\right)}}\*{{}_{2}F_{1}}\left({a,b\atop a+b-c+\frac{1}{2}(m+1)};\mathbf{I}% -\mathbf{T}\right),

\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}

;

{\frac{1}{2}}(j+1)-a\in\mathbb{N}

for some

j=1,\ldots,m

;

{\frac{1}{2}}(j+1)-c\notin\mathbb{N}

and

c-a-b-{\frac{1}{2}}(m-j)\notin\mathbb{N}

for all

j=1,\ldots,m

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $c$ : complex variable, $j$ : nonnegative integer, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Referenced by:: §35.7(ii), Erratum (V1.0.26) for Equation (35.7.8)
Permalink:: http://dlmf.nist.gov/35.7.E8
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.26):: Originally the constraint was written as $\Re\left(c\right),\Re\left(c-a-b\right)>\frac{1}{2}(m-1)$ . The constraint has been replaced with $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ ; ${\frac{1}{2}}(j+1)-a\in\mathbb{N}$ for some $j=1,\ldots,m$ ; ${\frac{1}{2}}(j+1)-c\notin\mathbb{N}$ and $c-a-b-{\frac{1}{2}}(m-j)\notin\mathbb{N}$ for all $j=1,\ldots,m$ ; for details see https://arxiv.org/abs/2002.05248.
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

…

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$\mathbb{N}$	set of all positive integers.
…

4: 27.12 Asymptotic Formulas: Primes

… ►A Carmichael number is a composite number

n

for which

b^{n}\equiv b\pmod{n}

for all

b\in\mathbb{N}

. …

5: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►Let

p

and

q

be nonnegative integers;

a_{1},\dots,a_{p}\in\mathbb{C}

;

b_{1},\dots,b_{q}\in\mathbb{C}

;

-b_{j}+\tfrac{1}{2}(k+1)\notin\mathbb{N}

1\leq j\leq q

1\leq k\leq m

. … ►If

-a_{j}+\tfrac{1}{2}(k+1)\in\mathbb{N}

for some

j, k

satisfying

1\leq j\leq p

1\leq k\leq m

, then the series expansion (35.8.1) terminates. …

6: 3.1 Arithmetics and Error Measures

… ►where

s

is equal to

1

0

, each

b_{j}

j\geq 1

, is either

0

1

b_{1}

is the most significant bit,

p

(

\in\mathbb{N}

) is the number of significant bits

b_{j}

b_{p-1}

is the least significant bit,

E

is an integer called the exponent,

b_{0}.b_{1}b_{2}\dots b_{p-1}

is the significand, and

f=.b_{1}b_{2}\dots b_{p-1}

is the fractional part. …

7: 18.36 Miscellaneous Polynomials

… ►The possibility of generalization to

\alpha=-k

, for

k\in\mathbb{N}

, is implicit in the identity Szegő (1975, page 102), …

8: 18.2 General Orthogonal Polynomials

… ►

\int_{a}^{b}|x|^{n}w(x)\,\mathrm{d}x<\infty,

n\in\mathbb{N}

…

Originally had the constraint $\Re\left(c\right),\Re\left(c-a-b\right)>\frac{1}{2}(m-1)$ . This constraint was replaced with $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ ; ${\frac{1}{2}}(j+1)-a\in\mathbb{N}$ for some $j=1,\ldots,m$ ; ${\frac{1}{2}}(j+1)-c\notin\mathbb{N}$ and $c-a-b-{\frac{1}{2}}(m-j)\notin\mathbb{N}$ for all $j=1,\ldots,m$ .

…