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21: 26.6 Other Lattice Path Numbers

… ►

Delannoy Number $D(m,n)$

► $D(m,n)$ 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 $(1,1)$. … ►

Table 26.6.4: Schröder numbers $r(n)$.

► ►►►

$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$	$n$	$r(n)$
$0$	$1$	$4$	$90$	$8$	$41586$	$12$	$\mathrm{272\hspace{0.33em}97738}$	$16$	$\mathrm{2\hspace{0.33em}09271\hspace{0.33em}56706}$
…

►

… ►

26.6.12 $$C\left(n\right)=\sum _{k=1}^{n}N(n,k),$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $k$: nonnegative integer, $n$: nonnegative integer and $N(n,k)$: Narayana number

Permalink:

http://dlmf.nist.gov/26.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iv), §26.6 and Ch.26

►

26.6.13 $$M(n)=\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)C\left(n+1-k\right),$$

ⓘ

Symbols:

$C\left(n\right)$: Catalan number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer and $M(n)$: Motzkin number

Permalink:

http://dlmf.nist.gov/26.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(iv), §26.6 and Ch.26

…

22: 26.13 Permutations: Cycle Notation

… ►An explicit representation of $\sigma$ can be given by the $2\times n$ matrix: … ►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. …In consequence, (26.13.2) can also be written as $\left(1,3,2,5,7\right)\left(6,8\right)$. … ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ … ►Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(2,3\right)\left(1,2\right)\left(4,5\right)\left(3,4\right)\left(2,3\right)\left(3,4\right)\left(4,5\right)\left(6,7\right)\left(5,6\right)\left(7,8\right)\left(6,7\right)$: $inv(\left(1,3,2,5,7\right)\left(6,8\right))=11$.

23: 33.20 Expansions for Small $|ϵ|$

… ►

$f(0,\mathrm{\ell };r)={(2r)}^{1/2}{J}_{2\mathrm{\ell }+1}\left(\sqrt{8r}\right),$

… ►where …The functions $J$ and $I$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by $C_{0,0}=1$, $C_{1,0}=0$, and … ►where $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and …The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6). …

24: 20.7 Identities

… ►

20.7.15 $$A\equiv A(\tau )=1/{\theta }_4\left(0\right|2\tau ),$$

ⓘ

Defines:

$A$ (locally)

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\equiv$: equals by definition and $\tau$: lattice parameter

Permalink:

http://dlmf.nist.gov/20.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

►

20.7.16 $${\theta }_1\left(2z\right|2\tau )=A{\theta }_1\left(z\right|\tau ){\theta }_2\left(z\right|\tau ),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $z$: complex, $\tau$: lattice parameter and $A$

Permalink:

http://dlmf.nist.gov/20.7.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

… ►

20.7.22 $${\theta }_2\left(4z\right|4\tau )=B{\theta }_2\left(\frac{1}{8}\pi -z\right|\tau ){\theta }_2\left(\frac{1}{8}\pi +z\right|\tau ){\theta }_2\left(\frac{3}{8}\pi -z\right|\tau ){\theta }_2\left(\frac{3}{8}\pi +z\right|\tau ),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex, $\tau$: lattice parameter and $B$

Permalink:

http://dlmf.nist.gov/20.7.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

►

20.7.23 $${\theta }_3\left(4z\right|4\tau )=B{\theta }_3\left(\frac{1}{8}\pi -z\right|\tau ){\theta }_3\left(\frac{1}{8}\pi +z\right|\tau ){\theta }_3\left(\frac{3}{8}\pi -z\right|\tau ){\theta }_3\left(\frac{3}{8}\pi +z\right|\tau ),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex, $\tau$: lattice parameter and $B$

Permalink:

http://dlmf.nist.gov/20.7.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(vi), §20.7 and Ch.20

… ►

20.7.34 $$\frac{{\theta }_1(z,q^2){\theta }_3(z,q^2)}{{\theta }_1(z,\mathrm{i}q)}=\frac{{\theta }_2(z,q^2){\theta }_4(z,q^2)}{{\theta }_2(z,\mathrm{i}q)}={\mathrm{i}}^{-1/4}\sqrt{\frac{{\theta }_2(0,q^2){\theta }_4(0,q^2)}{2}}.$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\mathrm{i}$: imaginary unit, $z$: complex and $q$: nome

Referenced by:

§20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/20.7.E34

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. See Walker (2012).

See also:

Annotations for §20.7(ix), §20.7 and Ch.20

…

25: 3.4 Differentiation

… ►where $A_k^n$ is as in (3.3.10). … ►If $f$ can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) …where $C$ is a simple closed contour described in the positive rotational sense such that $C$ and its interior lie in the domain of analyticity of $f$, and $x_0$ is interior to $C$. Taking $C$ to be a circle of radius $r$ centered at $x_0$, we obtain …The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2). …

26: 33.12 Asymptotic Expansions for Large $\eta$

… ►

$A_1=\frac{1}{5}{x}^2,$

►

$A_2=\frac{1}{35}(2x^3+6),$

►

$A_3=\frac{1}{15750}(21x^7+370x^4+580x),$

… ►For asymptotic expansions of $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ when $\eta \to \pm \mathrm{\infty }$ see Temme (2015, Chapter 31). … ►Then, by application of the results given in §§2.8(iii) and 2.8(iv), two sets of asymptotic expansions can be constructed for $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ when $\eta \to \mathrm{\infty }$. …

27: Bibliography N

… ►

Generalization of Binet’s Gamma function formulas. Integral Transforms Spec. Funct. 24 (8), pp. 597–606.

ⓘ

External Links:

ISSN 1065-2469, Document, FullText, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

BibTeX

… ►

Der Eulersche Dilogarithmus und seine Verallgemeinerungen. Nova Acta Leopoldina 90, pp. 123–212.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§25.12(i).

Encodings:

BibTeX

… ►

The asymptotic behavior of the general real solution of the third Painlevé equation. Dokl. Akad. Nauk SSSR 283 (5), pp. 1161–1165 (Russian).

ⓘ

Notes:

In Russian. English translation: Sov. Math., Dokl. 30(1988), no. 8, pp. 666–668.

External Links:

ISSN 0002-3264, MathReview (Vojislav Marić), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

… ►

ⓘ

Notes:

A large general purpose mathematical software collection containing some special function codes. Implementation is in single precision.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

►

High-frequency scattering by an impenetrable sphere. Ann. Physics 34 (1), pp. 23–95.

ⓘ

External Links:

Document, MathReview (E. Ar)

Referenced by:

§14.31(iii).

Encodings:

BibTeX

…

28: Bibliography O

… ►

Hyperterminants. II. J. Comput. Appl. Math. 89 (1), pp. 87–95.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§2.11(v).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

ⓘ

External Links:

ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

… ►

Connection formulas for second-order differential equations with multiple turning points. SIAM J. Math. Anal. 8 (1), pp. 127–154.

ⓘ

External Links:

Document, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v), (9.13.13), (9.13.17), §9.13(i), §9.13(i).

Encodings:

BibTeX

►

Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.

ⓘ

External Links:

Document, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v).

Encodings:

BibTeX

… ►

On an asymptotic expansion of a ratio of gamma functions. Proc. Roy. Irish Acad. Sect. A 95 (1), pp. 5–9.

ⓘ

External Links:

ISSN 0035-8975, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§5.11(iii).

Encodings:

BibTeX

…

29: DLMF Project News

error generating summary

30: 16.25 Methods of Computation

… ►There is, however, an added feature in the numerical solution of differential equations and difference equations (recurrence relations). …Instead a boundary-value problem needs to be formulated and solved. See §§3.6(vii), 3.7(iii), Olde Daalhuis and Olver (1998), Lozier (1980), and Wimp (1984, Chapters 7, 8).