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1: 26.21 Tables

… ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. …

2: 1.12 Continued Fractions

… ►If $C_n^{\prime }=C_{2n+1}$, $n=0,1,2,\mathrm{\dots }$, then $C^{\prime }$ is called the odd part of $C$. The odd part of $C$ exists iff $b_{2k+1}\ne 0$, $k=0,1,2,\mathrm{\dots }$, and up to equivalence is given by … ►and the even and odd parts of the continued fraction converge to finite values. …

3: 26.10 Integer Partitions: Other Restrictions

… ► $p(𝒪,n)$ denotes the number of partitions of $n$ into odd parts. …

4: 28.23 Expansions in Series of Bessel Functions

… ►When $j=2,3,4$ the series in the even-numbered equations converge for $\mathrm{\Re }z>0$ and $|\cosh z|>1$, and the series in the odd-numbered equations converge for $\mathrm{\Re }z>0$ and $|\sinh z|>1$. …

5: 18.17 Integrals

… ►

18.17.41 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}{\mathrm{𝐻𝑒}}_n\left(x\right)x^{z-1}dx=\mathrm{\Gamma }\left(z+n\right)a^{-n-2}{}_2{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}n,-\frac{1}{2}n+\frac{1}{2}}{-\frac{1}{2}z-\frac{1}{2}n,-\frac{1}{2}z-\frac{1}{2}n+\frac{1}{2}};-\frac{1}{2}{a}^2),$$ $\mathrm{\Re }a>0$. Also, $\mathrm{\Re }z>0$, $n$ even; $\mathrm{\Re }z>-1$, $n$ odd.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Laplace transform, Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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6: 9.13 Generalized Airy Functions

… ►

9.13.10

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Symbols:

$O\left(x\right)$: order not exceeding, $A_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (14)–(15))

Permalink:

http://dlmf.nist.gov/9.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

… ►

9.13.12

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Symbols:

$O\left(x\right)$: order not exceeding, $B_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\sin z$: sine function, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (17)–(18))

Permalink:

http://dlmf.nist.gov/9.13.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

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7: 23.15 Definitions

… ►In §§23.15–23.19, $k$ and $k^{\prime }$ $(\in ℂ)$ denote the Jacobi modulus and complementary modulus, respectively, and $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$ ($\mathrm{\Im }\tau >0$) denotes the nome; compare §§20.1 and 22.1. … ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$, ►

23.15.5 $$f(𝒜\tau )=c_{𝒜}{(c\tau +d)}^{\mathrm{\ell }}f(\tau ),$$ $\mathrm{\Im }\tau >0$,

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Symbols:

$\mathrm{\Im }$: imaginary part, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer, $f(\tau )$: modular function, $c_{𝒜}$: constant and $\mathrm{\ell }$: level

Permalink:

http://dlmf.nist.gov/23.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(i), §23.15 and Ch.23

►where $c_{𝒜}$ is a constant depending only on $𝒜$, and $\mathrm{\ell }$ (the level) is an integer or half an odd integer. …

8: 37.15 Orthogonal Polynomials on the Ball

… ►

37.15.21 $${𝒱}_{2n+1}^{\alpha }(𝐱;{𝔹}^d)=\underset{\mathit{ϵ}\in {\{0,1\}}^d;|\mathit{ϵ}|\mathrm{odd}}{⨁}{𝐱}^{\mathit{ϵ}}{𝒱}_{n-\frac{1}{2}(|\mathit{ϵ}|-1)}^{-\frac{\mathrm{𝟏}}{\mathrm{𝟐}}+\mathit{ϵ},\alpha }({𝐱}^2;{\mathrm{△}}^d).$$

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Symbols:

$\in$: element of, $n$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , $𝐱$: vector in $d$-dimensional Euclidean space, ${\mathrm{△}}^d$: simplex and ${𝔹}^d$: unit ball

Permalink:

http://dlmf.nist.gov/37.15.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §37.15(vii), §37.15, Part 37.III and Ch.37

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9: 20.2 Definitions and Periodic Properties

… ►For fixed $\tau$, each ${\theta }_j\left(z\right|\tau )$ is an entire function of $z$ with period $2\pi$; ${\theta }_1\left(z\right|\tau )$ is odd in $z$ and the others are even. For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$. …

10: 25.16 Mathematical Applications

… ► $H\left(s\right)$ is analytic for $\mathrm{\Re }s>1$, and can be extended meromorphically into the half-plane $\mathrm{\Re }s>-2k$ for every positive integer $k$ by use of the relations … ► $H\left(s\right)$ has a simple pole with residue $\zeta \left(1-2r\right)$ ($=-B_{2r}/(2r)$) at each odd negative integer $s=1-2r$, $r=1,2,3,\mathrm{\dots }$. … ►

25.16.11 $$H(s,z)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}\sum _{m=1}^n\frac{1}{m^z},$$ $\mathrm{\Re }\left(s+z\right)>1$,

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Symbols:

$H(s,z)$: generalized Euler sums, $\mathrm{\Re }$: real part, $m$: nonnegative integer, $n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

Euler sum, infinite series, series representation

Source:

Apostol and Vu (1984, (1), p. 86)

Permalink:

http://dlmf.nist.gov/25.16.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

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