DLMF: Untitled Document

… â–º

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.

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

denotes the number of partitions of

n

into parts with difference at least

k

. …If more than one restriction applies, then the restrictions are separated by commas, for example,

p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)

. … â–ºNote that

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

, with strict inequality for

n\geq 9

. …

… â–º

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}},

â“˜

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

… â–º

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

â–º

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

… â–º

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

.

â“˜

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

…

… â–ºLet

D

be a bounded domain with boundary

\partial D

and let

\overline{D}=D\cup\partial D

. … â–ºIf

u(z)

is harmonic in

D

,

z_{0}\in D

, and

u(z)\leq u(z_{0})

for all

z\in D

, then

u(z)

is constant in

D

. Moreover, if

D

is bounded and

u(z)

is continuous on

\overline{D}

and harmonic in

D

, then

u(z)

is maximum at some point on

\partial D

. … â–ºLet

F(z)

be a multivalued function and

D

be a domain. … â–ºSuppose

D

is a domain, and …

… â–ºThe complete case of

R_{J}

can be expressed in terms of

R_{F}

and

R_{D}

: … â–º

R_{D}\left(x,y,z\right)

is symmetric only in

x

and

y

, but either (nonzero)

x

or (nonzero)

y

can be moved to the third position by using … â–º

19.21.8

R_{D}\left(y,z,x\right)+R_{D}\left(z,x,y\right)+R_{D}\left(x,y,z\right)=3x^{-1% /2}y^{-1/2}z^{-1/2},

â“˜

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.21(ii), §19.33(iii), Erratum (V1.1.3) for Chapter 19
Permalink:: http://dlmf.nist.gov/19.21.E8
Encodings:: pMML, png, TeX
Correction (effective with 1.1.3):: The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.
Suggested 2021-06-07 by Luc Maisonobe
See also:: Annotations for §19.21(ii), §19.21 and Ch.19

… â–ºBecause

R_{G}

is completely symmetric,

x, y, z

can be permuted on the right-hand side of (19.21.10) so that

(x-z)(y-z)\leq 0

if the variables are real, thereby avoiding cancellations when

R_{G}

is calculated from

R_{F}

and

R_{D}

(see §19.36(i)). â–º

19.21.11

6R_{G}\left(x,y,z\right)=3(x+y+z)R_{F}\left(x,y,z\right)-\sum x^{2}R_{D}\left(% y,z,x\right)=\sum x(y+z)R_{D}\left(y,z,x\right),

â“˜

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.21(i), §19.21(ii)
Permalink:: http://dlmf.nist.gov/19.21.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §19.21(ii), §19.21 and Ch.19

…

… â–º

Delannoy Number $D(m,n)$

â–º

D(m,n)

is the number of paths from

(0,0)

to

(m,n)

that are composed of directed line segments of the form

(1,0)

,

(0,1)

, or

(1,1)

. â–º

26.6.1

D(m,n)=\sum_{k=0}^{n}\genfrac{(}{)}{0.0pt}{}{n}{k}\genfrac{(}{)}{0.0pt}{}{m+n-% k}{n}=\sum_{k=0}^{n}2^{k}\genfrac{(}{)}{0.0pt}{}{m}{k}\genfrac{(}{)}{0.0pt}{}{% n}{k}.

â“˜

Defines:: $D(m,n)$ : Delannoy number (locally)
Symbols:: $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : nonnegative integer, $m$ : nonnegative integer and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/26.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(i), §26.6(i), §26.6 and Ch.26

… â–º

26.6.5

\sum_{m,n=0}^{\infty}D(m,n)x^{m}y^{n}=\frac{1}{1-x-y-xy},

â“˜

Symbols:: $m$ : nonnegative integer, $n$ : nonnegative integer, $D(m,n)$ : Delannoy number, $x$ : small and $y$ : small
Permalink:: http://dlmf.nist.gov/26.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(ii), §26.6 and Ch.26

â–º

26.6.6

\sum_{n=0}^{\infty}D(n,n)x^{n}=\frac{1}{\sqrt{1-6x+x^{2}}},

â“˜

Symbols:: $n$ : nonnegative integer, $D(m,n)$ : Delannoy number and $x$ : small
Permalink:: http://dlmf.nist.gov/26.6.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §26.6(ii), §26.6 and Ch.26

…

… â–º

29.15.34

[D_{1},D_{3},\dots,D_{2n+1}]^{\mathrm{T}},

â“˜

Symbols:: $n$ : nonnegative integer and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E34
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

â–º

29.15.35

\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+1}^{2}+{\tfrac{1}{2}k^{2}% \left(\tfrac{1}{2}D_{1}^{2}-\sum_{p=0}^{n-1}D_{2p+1}D_{2p+3}\right)=1},

â“˜

Symbols:: $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : real parameter and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E35
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

â–º

29.15.36

\sum_{p=0}^{n}(2p+1)D_{2p+1}>0.

â“˜

Symbols:: $n$ : nonnegative integer, $p$ : nonnegative integer and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E36
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

… â–º

29.15.40

\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{n}D_{2p+2}^{2}-\tfrac{1}{2}k^{2}% \sum_{p=1}^{n}D_{2p}D_{2p+2}=1,

â“˜

Symbols:: $n$ : nonnegative integer, $p$ : nonnegative integer, $k$ : real parameter and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E40
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

â–º

29.15.41

\sum_{p=0}^{n}(2p+2)D_{2p+2}>0.

â“˜

Symbols:: $n$ : nonnegative integer, $p$ : nonnegative integer and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.15.E41
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

…

… â–ºWe denote it by

\mathcal{D}(I)

. … â–ºfor all

\phi\in\mathcal{D}(I)

. … â–ºFor any locally integrable function

f

, its distributional derivative is

Df=\Lambda^{\prime}_{f}

. … â–ºThe distributional derivative

D^{k}f

of

f

is defined by …

… â–º

29.6.42

\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=0}^{\infty}D_{2p+1}^{2}+{\tfrac{1}{2}k% ^{2}\left(\tfrac{1}{2}D_{1}^{2}-\sum_{p=0}^{\infty}D_{2p+1}D_{2p+3}\right)=1},

â“˜

Symbols:: $p$ : nonnegative integer, $k$ : real parameter and $D_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E42
Encodings:: pMML, png, TeX
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

â–º

29.6.43

\sum_{p=0}^{\infty}(2p+1)D_{2p+1}>0,

â“˜

Symbols:: $p$ : nonnegative integer and $D_{2p+1}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E43
Encodings:: pMML, png, TeX
See also:: Annotations for §29.6(iii), §29.6 and Ch.29

… â–º

29.6.53

\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \sum_{p=1}^{\infty}D_{2p}\sin\left(2p\phi\right),

â“˜

Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathit{Es}^{\NVar{m}}_{\NVar{\nu}}\left(\NVar{z},\NVar{k^{2}}\right)$ : Lamé function, $\sin\NVar{z}$ : sine function, $m$ : nonnegative integer, $p$ : nonnegative integer, $z$ : complex variable, $k$ : real parameter, $\nu$ : real parameter, $\phi$ : change of variable and $D_{2p}$ : coefficents
Referenced by:: §29.15(i)
Permalink:: http://dlmf.nist.gov/29.6.E53
Encodings:: pMML, png, TeX
See also:: Annotations for §29.6(iv), §29.6 and Ch.29

… â–º

29.6.57

\left(1-\tfrac{1}{2}k^{2}\right)\sum_{p=1}^{\infty}D_{2p}^{2}-\tfrac{1}{2}k^{2% }\sum_{p=1}^{\infty}D_{2p}D_{2p+2}=1,

â“˜

Symbols:: $p$ : nonnegative integer, $k$ : real parameter and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E57
Encodings:: pMML, png, TeX
See also:: Annotations for §29.6(iv), §29.6 and Ch.29

â–º

29.6.58

\sum_{p=0}^{\infty}(2p+2)D_{2p+2}>0,

â“˜

Symbols:: $p$ : nonnegative integer and $D_{2p}$ : coefficents
Permalink:: http://dlmf.nist.gov/29.6.E58
Encodings:: pMML, png, TeX
See also:: Annotations for §29.6(iv), §29.6 and Ch.29

…

… â–º

D=b^{2}-4ac.

â–ºThe sum and product of the roots are respectively

-b/a

and

c/a

. … â–ºLet â–º

D_{1}=a_{1},

… â–ºThen

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

.

… â–ºIn the following formulas for the coefficients

A_{k}(\zeta)

,

B_{k}(\zeta)

,

C_{k}(\zeta)

, and

D_{k}(\zeta)

,

u_{k}

,

v_{k}

are the constants defined in §9.7(i), and

U_{k}(p)

,

V_{k}(p)

are the polynomials in

p

of degree

3k

defined in §10.41(ii). … â–ºNote: Another way of arranging the above formulas for the coefficients

A_{k}(\zeta),B_{k}(\zeta),C_{k}(\zeta)

, and

D_{k}(\zeta)

would be by analogy with (12.10.42) and (12.10.46). … â–ºEach of the coefficients

A_{k}(\zeta)

,

B_{k}(\zeta)

,

C_{k}(\zeta)

, and

D_{k}(\zeta)

,

k=0,1,2,\dotsc

, is real and infinitely differentiable on the interval

-\infty<\zeta<\infty

. … â–ºFor numerical tables of

\zeta=\zeta(z)

,

(4\zeta/(1-z^{2}))^{\frac{1}{4}}

and

A_{k}(\zeta)

,

B_{k}(\zeta)

,

C_{k}(\zeta)

, and

D_{k}(\zeta)

see Olver (1962, pp. 28–42). … â–ºThe equations of the curved boundaries

D_{1}E_{1}

and

D_{2}E_{2}

in the

\zeta

-plane are given parametrically by …

DÅ¾rbasjan sum

1—10 of 407 matching pages

1: 26.10 Integer Partitions: Other Restrictions

2: 28.25 Asymptotic Expansions for Large $\Re z$

3: 1.10 Functions of a Complex Variable

4: 19.21 Connection Formulas

5: 26.6 Other Lattice Path Numbers

Delannoy Number $D(m,n)$

6: 29.15 Fourier Series and Chebyshev Series

7: 1.16 Distributions

8: 29.6 Fourier Series

9: 1.11 Zeros of Polynomials

10: 10.20 Uniform Asymptotic Expansions for Large Order

DÅ¾rbasjan sum

1—10 of 407 matching pages

1: 26.10 Integer Partitions: Other Restrictions

2: 28.25 Asymptotic Expansions for Large ℜ â¡ z

3: 1.10 Functions of a Complex Variable

4: 19.21 Connection Formulas

5: 26.6 Other Lattice Path Numbers

Delannoy Number D â¡ ( m , n )

6: 29.15 Fourier Series and Chebyshev Series

7: 1.16 Distributions

8: 29.6 Fourier Series

9: 1.11 Zeros of Polynomials

10: 10.20 Uniform Asymptotic Expansions for Large Order

2: 28.25 Asymptotic Expansions for Large $\Re z$

Delannoy Number $D(m,n)$