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11: 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

. … ►The number of plane partitions of

n

is denoted by

\operatorname{pp}\left(n\right)

, with

\operatorname{pp}\left(0\right)=1

. … ►in

B(2r+1,2s,2t)

it is …in

B(2r+1,2s+1,2t)

it is … ►in

B(2r+1,2r+1,2t)

it is …

12: 18.8 Differential Equations

… ►

Table 18.8.1: Classical OP’s: differential equations

A(x)f^{\prime\prime}(x)+B(x)f^{\prime}(x)+C(x)f(x)+\lambda_{n}f(x)=0

► ►►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
…
7	$P_{n}\left(x\right)$	$1-x^{2}$	$-2x$	$0$	$n(n+1)$
…
10	${\mathrm{e}}^{-\frac{1}{2}x}x^{\frac{1}{2}\alpha}L^{(\alpha)}_{n}\left(x\right)$	$x$	$1$	$-\frac{1}{4}x-\frac{1}{4}\alpha^{2}x^{-1}$	$n+\frac{1}{2}(\alpha+1)$
11	${\mathrm{e}}^{-n^{-1}x}x^{\ell+1}L^{(2\ell+1)}_{n-\ell-1}\left(2n^{-1}x\right)$	$1$	$0$	$\frac{2}{x}-\frac{\ell(\ell+1)}{x^{2}}$	$-\frac{1}{n^{2}}$
12	$H_{n}\left(x\right)$	$1$	$-2x$	$0$	$2n$
…

► ►Item 11 of Table 18.8.1 yields (18.39.36) for

Z=1

13: 34.2 Definition: $\mathit{3j}$ Symbol

… ►The quantities

j_{1},j_{2},j_{3}

in the

\mathit{3j}

symbol are called angular momenta. …where

r, s, t

is any permutation of

1,2,3

. The corresponding projective quantum numbers

m_{1},m_{2},m_{3}

are given by … ►

See accompanying text — Figure 34.2.1: Angular momenta $j_{r}$ and projective quantum numbers $m_{r}$ , $r=1,2,3$ . Magnify

… ►

34.2.4

\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&m_{3}\end{pmatrix}={(-1)^{j_{1}-j_{2}-m_{3}}}\Delta(j_{1}j_{2}j_{3% })\left((j_{1}+m_{1})!(j_{1}-m_{1})!(j_{2}+m_{2})!(j_{2}-m_{2})!(j_{3}+m_{3})!% (j_{3}-m_{3})!\right)^{\frac{1}{2}}\*\sum_{s}\frac{(-1)^{s}}{s!(j_{1}+j_{2}-j_% {3}-s)!(j_{1}-m_{1}-s)!(j_{2}+m_{2}-s)!(j_{3}-j_{2}+m_{1}+s)!(j_{3}-j_{1}-m_{2% }+s)!},

ⓘ

Defines:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol
Symbols:: $!$ : factorial (as in $n!$ ), $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $\Delta(j_{1}j_{2}j_{3})$ : product and $s$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/34.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §34.2 and Ch.34

…

14: 24.12 Zeros

… ►In the interval

0\leq x\leq 1

the only zeros of

B_{2n+1}\left(x\right)

n=1,2,\dotsc

, are

0,\tfrac{1}{2},1

, and the only zeros of

B_{2n}\left(x\right)-B_{2n}

n=1,2,\dotsc

, are

0,1

. … ►Then the zeros in the interval

-\infty<x\leq\frac{1}{2}

are

1-x_{j}^{(n)}

. … ►When

n

is odd

x^{(n)}_{1}=\frac{1}{2}

x^{(n)}_{2}=1

(n\geq 3)

, and as

n\to\infty

with

m(\geq 1)

fixed, … ►Then

R(n)=n

when

1\leq n\leq 5

, and … ►When

n

is odd

y^{(n)}_{1}=\tfrac{1}{2}

, …

15: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

… ►Jacobian elliptic functions with real moduli in the intervals

(-\infty,0)

and

(1,\infty)

, or with purely imaginary moduli are related to functions with moduli in the interval

[0,1]

by the following formulas. … ►

k_{1}k_{1}^{\prime}=\frac{k}{1+k^{2}},

… ►In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of

k

, irrespective of which values of

\sqrt{k}

and

k^{\prime}=\sqrt{1-k^{2}}

are chosen—as long as they are used consistently. For proofs of these results and further information see Walker (2003).

16: 26.16 Multiset Permutations

… ►Let

S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}

be the multiset that has

a_{j}

copies of

j

1\leq j\leq n

. …The number of elements in

\mathfrak{S}_{S}

is the multinomial coefficient (§26.4)

\genfrac{(}{)}{0.0pt}{}{a_{1}+a_{2}+\cdots+a_{n}}{a_{1},a_{2},\ldots,a_{n}}

. … ►The definitions of inversion number and major index can be extended to permutations of a multiset such as

351322453154\in\mathfrak{S}_{\{1^{2},2^{2},3^{3},4^{2},5^{3}\}}

. Thus

\mathop{\mathrm{inv}}(351322453154)=4+8+0+3+1+1+2+3+1+0+1=24

, and

\mathop{\mathrm{maj}}(351322453154)=2+4+8+9+11=34.

… ►and again with

S=\{1^{a_{1}},2^{a_{2}},\ldots,n^{a_{n}}\}

we have …

17: 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. … ►If

p=q+1

, then (35.8.1) converges absolutely for

\|\mathbf{T}\|<1

and diverges for

\|\mathbf{T}\|>1

. ►If

p>q+1

, then (35.8.1) diverges unless it terminates. … ►Let

a_{1}+a_{2}+a_{3}+\frac{1}{2}(m+1)=b_{1}+b_{2}

; one of the

a_{j}

be a negative integer;

\Re\left(b_{1}-a_{1}\right)

\Re\left(b_{1}-a_{2}\right)

\Re\left(b_{1}-a_{3}\right)

\Re\left(b_{1}-a_{1}-a_{2}-a_{3}\right)>\frac{1}{2}(m-1)

. …

18: 15.8 Transformations of Variable

… ►The hypergeometric functions that correspond to Groups 1 and 2 have

z

as variable. … ►

Group 1 $\longrightarrow$ Group 3

… ►

Group 2 $\longrightarrow$ Group 1

… ►

b=\tfrac{1}{3}a+\tfrac{1}{3}

c=2b=a-b+1

in Groups 1 and 2. … ►This is a quadratic transformation between two cases in Group 1. …

19: 26.6 Other Lattice Path Numbers

… ►

D(m,n)

is the number of paths from

(0,0)

(m,n)

that are composed of directed line segments of the form

(1,0)

(0,1)

, or

(1,1)

. … ►

M(n)

is the number of lattice paths from

(0,0)

(n,n)

that stay on or above the line

y=x

and are composed of directed line segments of the form

(2,0)

(0,2)

, or

(1,1)

. … ►

N(n,k)

is the number of lattice paths from

(0,0)

(n,n)

that stay on or above the line

y=x

, are composed of directed line segments of the form

(1,0)

(0,1)

, and for which there are exactly

k

occurrences at which a segment of the form

(0,1)

is followed by a segment of the form

(1,0)

. … ►

r(n)

is the number of paths from

(0,0)

(n,n)

that stay on or above the diagonal

y=x

and are composed of directed line segments of the form

(1,0)

(0,1)

, or

(1,1)

. … ►

§26.6(iv) Identities

…

20: 26.11 Integer Partitions: Compositions

… ►For example, there are eight compositions of 4:

4,3+1,1+3,2+2,2+1+1,1+2+1,1+1+2

, and

1+1+1+1

c\left(n\right)

denotes the number of compositions of

n

, and

c_{m}\left(n\right)

is the number of compositions into exactly

m

parts.

c\left(\in\!T,n\right)

is the number of compositions of

n

with no 1’s, where again

T=\{2,3,4,\ldots\}

. … ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).

👉1(716)351-6210📞Delta Flights 🪐Airlines 🛸Ticket Change Phone 🌍Number

11—20 of 845 matching pages

11: 26.12 Plane Partitions

12: 18.8 Differential Equations

13: 34.2 Definition: 3 ⁢ j Symbol

14: 24.12 Zeros

15: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

16: 26.16 Multiset Permutations

17: 35.8 Generalized Hypergeometric Functions of Matrix Argument

18: 15.8 Transformations of Variable

Group 1 ⟶ Group 3

Group 2 ⟶ Group 1

19: 26.6 Other Lattice Path Numbers

§26.6(iv) Identities

20: 26.11 Integer Partitions: Compositions

13: 34.2 Definition: $\mathit{3j}$ Symbol

Group 1 $\longrightarrow$ Group 3

Group 2 $\longrightarrow$ Group 1