symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments. …

2: 26.10 Integer Partitions: Other Restrictions

… ►

p\left(\mathcal{D},n\right)

denotes the number of partitions of

n

into distinct parts.

p_{m}\left(\mathcal{D},n\right)

denotes the number of partitions of

n

into at most

m

distinct parts. … ►

Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from

A_{j,k}

► ►►►

	$p\left(\mathcal{D},n\right)$	$p\left(\mathcal{D}2,n\right)$	$p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)$	$p\left(\mathcal{D}^{\prime}3,n\right)$
…
$9$	$8$	$5$	$3$	$3$
…

► … ►Note that

p\left(\mathcal{D}^{\prime}3,n\right)\leq p\left(\mathcal{D}3,n\right)

, with strict inequality for

n\geq 9

. It is known that for

k>3

p\left(\mathcal{D}k,n\right)\geq p\left(\in\!A_{1,k+3},n\right)

, with strict inequality for

n

sufficiently large, provided that

k=2^{m}-1,m=3,4,5

, or

k\geq 32

; see Yee (2004). …

3: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

►

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)

. … ►

Table 26.6.1: Delannoy numbers

D(m,n)

► ►►►

$m$	$n$
$m$	0	1	2	3	4	5	6	7	8	9	10
…

► … ►

Table 26.6.2: Motzkin numbers

M(n)

► ►►►

$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$	$n$	$M(n)$
0	1	4	9	8	323	12	15511	16	8 53467
…

► … ►

Table 26.6.3: Narayana numbers

N(n,k)

► ►►►

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
…

► …

4: 1.11 Zeros of Polynomials

… ►Set

z=w-\tfrac{1}{3}a

to reduce

f(z)=z^{3}+az^{2}+bz+c

g(w)=w^{3}+pw+q

, with

p=(3b-a^{2})/3

q=(2a^{3}-9ab+27c)/27

. … ►

f(z)=z^{3}-6z^{2}+6z-2

g(w)=w^{3}-6w-6

A=3\sqrt[3]{4}

B=3\sqrt[3]{2}

. … ►Resolvent cubic is

z^{3}+12z^{2}+20z+9=0

with roots

\theta_{1}=-1

\theta_{2}=-\tfrac{1}{2}(11+\sqrt{85})

\theta_{3}=-\tfrac{1}{2}(11-\sqrt{85})

, and

\sqrt{-\theta_{1}}=1

\sqrt{-\theta_{2}}=\tfrac{1}{2}(\sqrt{17}+\sqrt{5})

\sqrt{-\theta_{3}}=\tfrac{1}{2}(\sqrt{17}-\sqrt{5})

. … ►Let … ►Then

f(z)

, with

a_{n}\not=0

, is stable iff

a_{0}\not=0

;

D_{2k}>0

k=1,\dots,\left\lfloor\frac{1}{2}n\right\rfloor

;

\operatorname{sign}D_{2k+1}=\operatorname{sign}a_{0}

k=0,1,\dots,\left\lfloor\frac{1}{2}n-\frac{1}{2}\right\rfloor

5: 3.3 Interpolation

… ►If

f

is analytic in a simply-connected domain

D

(§1.13(i)), then for

z\in D

, …where

C

is a simple closed contour in

D

described in the positive rotational sense and enclosing the points

z,z_{1},z_{2},\dots,z_{n}

. … ►If

f

is analytic in a simply-connected domain

D

, then for

z\in{D}

, …where

\omega_{n+1}(\zeta)

is given by (3.3.3), and

C

is a simple closed contour in

{D}

described in the positive rotational sense and enclosing

z_{0},z_{1},\dots,z_{n}

. … ►Then by using

x_{3}

in Newton’s interpolation formula, evaluating

\left[x_{0},x_{1},x_{2},x_{3}\right]f=-0.26608\;28233

and recomputing

f^{\prime}(x)

, another application of Newton’s rule with starting value

x_{3}

gives the approximation

x=2.33810\;7373

, with 8 correct digits. …

6: 21.5 Modular Transformations

… ►Let

\mathbf{A}

\mathbf{B}

\mathbf{C}

, and

\mathbf{D}

g\times g

matrices with integer elements such that ►

21.5.1

\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{C}&\mathbf{D}\end{bmatrix}

ⓘ

Defines:: Matrix $\boldsymbol{{\Gamma}}$ (locally)
Permalink:: http://dlmf.nist.gov/21.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §21.5(i), §21.5 and Ch.21

… ►Here

\xi(\boldsymbol{{\Gamma}})

is an eighth root of unity, that is,

(\xi(\boldsymbol{{\Gamma}}))^{8}=1

. … ►(

\mathbf{A}

invertible with integer elements.) …For a

g\times g

matrix

{\bf A}

we define

\operatorname{diag}\mathbf{A}

, as a column vector with the diagonal entries as elements. …

7: 27.2 Functions

… ►It is the special case

k=2

of the function

d_{k}\left(n\right)

that counts the number of ways of expressing

n

as the product of

k

factors, with the order of factors taken into account. …Note that

\sigma_{0}\left(n\right)=d\left(n\right)

. … ►Table 27.2.2 tabulates the Euler totient function

\phi\left(n\right)

, the divisor function

d\left(n\right)

(

=\sigma_{0}(n)

), and the sum of the divisors

\sigma(n)

(

=\sigma_{1}(n)

), for

n=1(1)52

. … ►

Table 27.2.2: Functions related to division.

► ►►►►

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

►

8: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

…

9: 28.25 Asymptotic Expansions for Large $\Re z$

… ►

28.25.1

{\operatorname{M}^{(3,4)}_{\nu}}\left(z,h\right)\sim\frac{e^{\pm\mathrm{i}% \left(2h\cosh z-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi\right)}}{\left(\pi h% (\cosh z+1)\right)^{\frac{1}{2}}}\*\sum_{m=0}^{\infty}\dfrac{D^{\pm}_{m}}{% \left(\mp 4\mathrm{i}h(\cosh z+1)\right)^{m}},

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable, $\nu$ : complex parameter and $D_{m}^{\pm}$ : coefficients
A&S Ref:: 20.9.1 (for 3 and for integer $\nu$ ) 20.9.3 (for 4 and for integer $\nu$ )
Referenced by:: §28.25
Permalink:: http://dlmf.nist.gov/28.25.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.25 and Ch.28

… ►

D_{-1}^{\pm}=0,

►

D_{0}^{\pm}=1,

… ►

28.25.3

(m+1)D^{\pm}_{m+1}+{\left((m+\tfrac{1}{2})^{2}\pm(m+\tfrac{1}{4})8\mathrm{i}h+% 2h^{2}-a\right)D^{\pm}_{m}}\pm(m-\tfrac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1% }^{\pm}=0,

m\geq 0

ⓘ

Symbols:: $\mathrm{i}$ : imaginary unit, $m$ : integer, $h$ : parameter, $a$ : parameter and $D_{m}^{\pm}$ : coefficients
A&S Ref:: 20.9.2 (for $+$ ) 20.9.4 (for $-$ )
Permalink:: http://dlmf.nist.gov/28.25.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §28.25 and Ch.28

…

10: 28.8 Asymptotic Expansions for Large $q$

… ►

28.8.1

\rselection{a_{m}\left(h^{2}\right)\\ b_{m+1}\left(h^{2}\right)}\sim-2h^{2}+2sh-\frac{1}{8}(s^{2}+1)-\frac{1}{2^{7}h% }(s^{3}+3s)-\frac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\frac{1}{2^{17}h^{3}}(33s^% {5}+410s^{3}+405s)-\frac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}+486)-% \frac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.

ⓘ

Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\sim$ : Poincaré asymptotic expansion, $m$ : integer and $h$ : parameter
A&S Ref:: 20.2.30 (in different form)
Referenced by:: §28.8(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)
Permalink:: http://dlmf.nist.gov/28.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(i), §28.8 and Ch.28

… ►Also let

\xi=2\sqrt{h}\cos x

and

D_{m}\left(\xi\right)=e^{-\ifrac{\xi^{2}}{4}}\mathit{He}_{m}\left(\xi\right)

(§18.3). … ►

28.8.4

U_{m}(\xi)\sim D_{m}\left(\xi\right)-\frac{1}{2^{6}h}\left(D_{m+4}\left(\xi% \right)-4!\dbinom{m}{4}D_{m-4}\left(\xi\right)\right)+\frac{1}{2^{13}h^{2}}% \left(D_{m+8}\left(\xi\right)-2^{5}(m+2)D_{m+4}\left(\xi\right)+4!\,2^{5}(m-1)% \dbinom{m}{4}D_{m-4}\left(\xi\right)+8!\genfrac{(}{)}{0.0pt}{}{m}{8}D_{m-8}% \left(\xi\right)\right)+\cdots,

ⓘ

Defines:: $U_{m}(\xi)$ : function (locally)
Symbols:: $D_{\NVar{\nu}}\left(\NVar{z}\right)$ : parabolic cylinder function, $\sim$ : Poincaré asymptotic expansion, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $m$ : integer, $h$ : parameter and $\xi$ : variable
A&S Ref:: 20.9.17 (in different form)
Permalink:: http://dlmf.nist.gov/28.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(ii), §28.8 and Ch.28

►

28.8.5

V_{m}(\xi)\sim\frac{1}{2^{4}h}\bigg{(}-D_{m+2}\left(\xi\right)-m(m-1)D_{m-2}% \left(\xi\right)\bigg{)}+\frac{1}{2^{10}h^{2}}\left(D_{m+6}\left(\xi\right)+(m% ^{2}-25m-36)D_{m+2}\left(\xi\right)-m(m-1)(m^{2}+27m-10)D_{m-2}\left(\xi\right% )-6!\genfrac{(}{)}{0.0pt}{}{m}{6}D_{m-6}\left(\xi\right)\right)+\cdots,

ⓘ

Defines:: $V_{m}(\xi)$ : function (locally)
Symbols:: $D_{\NVar{\nu}}\left(\NVar{z}\right)$ : parabolic cylinder function, $\sim$ : Poincaré asymptotic expansion, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $m$ : integer, $h$ : parameter and $\xi$ : variable
A&S Ref:: 20.9.18 (in slightly different form)
Referenced by:: Erratum (V1.0.16) for Equation (28.8.5)
Permalink:: http://dlmf.nist.gov/28.8.E5
Encodings:: pMML, png, TeX
Error (effective with 1.0.16):: Originally the $-$ in front of the $6!$ was given incorrectly as $+$ .
Suggested 2017-02-02 by Daniel Karlsson
See also:: Annotations for §28.8(ii), §28.8 and Ch.28

… ►

28.8.6

\widehat{C}_{m}\sim\left(\frac{\pi h}{2(m!)^{2}}\right)^{\ifrac{1}{4}}\left(1+% \frac{2m+1}{8h}+\dfrac{m^{4}+2m^{3}+263m^{2}+262m+108}{2048h^{2}}+\cdots\right% )^{-\ifrac{1}{2}},

ⓘ

Defines:: $\widehat{C}_{m}$ : coefficient (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $m$ : integer and $h$ : parameter
A&S Ref:: 20.9.19 (in slightly different form)
Permalink:: http://dlmf.nist.gov/28.8.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(ii), §28.8 and Ch.28

…

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…

1—10 of 603 matching pages

1: 34.6 Definition: 9 ⁢ j Symbol

§34.6 Definition: 9 ⁢ j Symbol

2: 26.10 Integer Partitions: Other Restrictions

3: 26.6 Other Lattice Path Numbers

Delannoy Number D ⁡ ( m , n )

4: 1.11 Zeros of Polynomials

5: 3.3 Interpolation

6: 21.5 Modular Transformations

7: 27.2 Functions

8: 8 Incomplete Gamma and Related Functions

Chapter 8 Incomplete Gamma and Related Functions

9: 28.25 Asymptotic Expansions for Large ℜ ⁡ z

10: 28.8 Asymptotic Expansions for Large q

1: 34.6 Definition: $\mathit{9j}$ Symbol

§34.6 Definition: $\mathit{9j}$ Symbol

Delannoy Number $D(m,n)$

8: 8 Incomplete Gamma and Related
Functions

9: 28.25 Asymptotic Expansions for Large $\Re z$

10: 28.8 Asymptotic Expansions for Large $q$

$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$	$n$	$\phi\left(n\right)$	$d\left(n\right)$	$\sigma\left(n\right)$
…
3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
…
11	10	2	12	24	8	8	60	37	36	2	38	50	20	6	93
…