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31: 26.6 Other Lattice Path Numbers

§26.6 Other Lattice Path Numbers

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Delannoy Number $D(m,n)$

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Motzkin Number $M(n)$

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Narayana Number $N(n,k)$

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Schröder Number $r(n)$

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32: 29.2 Differential Equations

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29.2.9 $$\frac{d^{2}w}{{d\eta }^2}+(g-\nu (\nu +1)\mathrm{\wp }\left(\eta \right))w=0,$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\nu$: real parameter, $g$ and $\eta$

Permalink:

http://dlmf.nist.gov/29.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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29.2.11 $$\zeta =\mathrm{\wp }(\eta ;g_2,g_3)=\mathrm{\wp }\left(\eta \right),$$

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Defines:

$\zeta$ (locally)

Symbols:

$\mathrm{\wp }(z;g_2,g_3)$: Weierstrass $\mathrm{\wp }$-function, $\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\eta$ , $g_2$ and $g_3$

Permalink:

http://dlmf.nist.gov/29.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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33: 20.7 Identities

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

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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

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

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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

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§20.7(viii) Transformations of Lattice Parameter

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20.7.28 $${\theta }_3\left(z\right|\tau +1)={\theta }_4\left(z\right|\tau ),$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/20.7.E28

Encodings:

pMML, png, TeX

See also:

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

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20.7.29 $${\theta }_4\left(z\right|\tau +1)={\theta }_3\left(z\right|\tau ).$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/20.7.E29

Encodings:

pMML, png, TeX

See also:

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

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34: 20.10 Integrals

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§20.10(i) Mellin Transforms with respect to the Lattice Parameter

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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

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35: 20.1 Special Notation

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$m$, $n$	integers.
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$\tau$ $(\in ℂ)$	the lattice parameter, $\mathrm{\Im }\tau >0$.
$q$ $(\in ℂ)$	the nome, $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, . Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\mathrm{\Re }\tau =0$, $\mathrm{\Im }\tau >0$, so that and $\tau$ and $q$ are uniquely related.
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36: 26.9 Integer Partitions: Restricted Number and Part Size

… ►It follows that $p_k\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$. … ►It is also equal to the number of lattice paths from $(0,0)$ to $(m,k)$ that have exactly $n$ vertices $(h,j)$, $1\le h\le m$, $1\le j\le k$, above and to the left of the lattice path. … ►

Figure 26.9.2: The partition $5+5+3+2$ represented as a lattice path.

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37: 20.5 Infinite Products and Related Results

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20.5.5 $${\theta }_1\left(z\right|\tau )={\theta }_1^{\prime }\left(0\right|\tau )\sin z\prod _{n=1}^{\mathrm{\infty }}\frac{\sin \left(n\pi \tau +z\right)\sin \left(n\pi \tau -z\right)}{{\sin }^2\left(n\pi \tau \right)},$$

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Symbols:

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

Referenced by:

§20.5(i), §20.5(iii), §20.7(iv)

Permalink:

http://dlmf.nist.gov/20.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

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20.5.6 $${\theta }_2\left(z\right|\tau )={\theta }_2\left(0\right|\tau )\cos z\prod _{n=1}^{\mathrm{\infty }}\frac{\cos \left(n\pi \tau +z\right)\cos \left(n\pi \tau -z\right)}{{\cos }^2\left(n\pi \tau \right)},$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/20.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

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20.5.7 $${\theta }_3\left(z\right|\tau )={\theta }_3\left(0\right|\tau )\prod _{n=1}^{\mathrm{\infty }}\frac{\cos \left((n-\frac{1}{2})\pi \tau +z\right)\cos \left((n-\frac{1}{2})\pi \tau -z\right)}{{\cos }^2\left((n-\frac{1}{2})\pi \tau \right)},$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/20.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

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20.5.8 $${\theta }_4\left(z\right|\tau )={\theta }_4\left(0\right|\tau )\prod _{n=1}^{\mathrm{\infty }}\frac{\sin \left((n-\frac{1}{2})\pi \tau +z\right)\sin \left((n-\frac{1}{2})\pi \tau -z\right)}{{\sin }^2\left((n-\frac{1}{2})\pi \tau \right)}.$$

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Symbols:

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

Referenced by:

§20.5(iii), §20.7(iv)

Permalink:

http://dlmf.nist.gov/20.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

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20.5.9 $${\theta }_3\left(\pi z\right|\tau )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{p}^{2n}{q}^{n^2}=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+q^{2n-1}{p}^2\right)\left(1+q^{2n-1}{p}^{-2}\right),$$

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Symbols:

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

Referenced by:

§17.8, §20.12(i), §20.5(i)

Permalink:

http://dlmf.nist.gov/20.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5(i), §20.5 and Ch.20

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38: 21.3 Symmetry and Quasi-Periodicity

… ►The set of points ${𝐦}_1+𝛀{𝐦}_2$ form a $g$-dimensional lattice, the period lattice of the Riemann theta function. …

39: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

§26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

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§26.4(i) Definitions

… ►It is also the number of $k$-dimensional lattice paths from $(0,0,\mathrm{\dots },0)$ to $(n_1,n_2,\mathrm{\dots },n_k)$. …

40: 20.9 Relations to Other Functions

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20.9.1 $$k={\theta }_2^2\left(0\right|\tau )/{\theta }_3^2\left(0\right|\tau )$$

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Defines:

$k$: modulus (locally)

Symbols:

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

Referenced by:

§20.11(iii), §20.9(ii), §23.6(ii)

Permalink:

http://dlmf.nist.gov/20.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.9(i), §20.9 and Ch.20

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