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11: 26.17 The Twelvefold Way

… ►The twelvefold way gives the number of mappings $f$ from set $N$ of $n$ objects to set $K$ of $k$ objects (putting balls from set $N$ into boxes in set $K$). …In this table ${\left(k\right)}_n$ is Pochhammer’s symbol, and $S(n,k)$ and $p_k\left(n\right)$ are defined in §§26.8(i) and 26.9(i). …

12: 4.28 Definitions and Periodicity

… ►The zeros of $\sinh z$ and $\cosh z$ are $z=\mathrm{i}k\pi$ and $z=\mathrm{i}\left(k+\frac{1}{2}\right)\pi$, respectively, $k\in \mathbb{Z}$.

13: 5.1 Special Notation

… ► ►►►

$j,m,n$	nonnegative integers.
$k$	nonnegative integer, except in §5.20.
…

…

14: 22.2 Definitions

… ►The nome $q$ is given in terms of the modulus $k$ by …where $K\left(k\right)$, $K^{\prime }\left(k\right)$ are defined in §19.2(ii). …where $k^{\prime }=\sqrt{1-k^2}$ and the theta functions are defined in §20.2(i). … ►Each is meromorphic in $z$ for fixed $k$, with simple poles and simple zeros, and each is meromorphic in $k$ for fixed $z$. For $k\in [0,1]$, all functions are real for $z\in ℝ$. …

15: 10.14 Inequalities; Monotonicity

… ►

10.14.9 $$|J_n\left(nz\right)|\le 1,$$ $n=0,1,2,\mathrm{\dots },z\in 𝐊$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\in$: element of, $n$: integer, $z$: complex variable and $𝐊$: domain

Permalink:

http://dlmf.nist.gov/10.14.E9

Encodings:

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See also:

Annotations for §10.14, §10.14 and Ch.10

►where $𝐊$ is defined in §10.20(ii). …

16: 18.36 Miscellaneous Polynomials

… ►The possibility of generalization to $\alpha =-k$, for $k\in \mathbb{N}$, is implicit in the identity Szegő (1975, page 102), …implying that, for $n\ge k$, the orthogonality of the $L_n^{(-k)}\left(x\right)$ with respect to the Laguerre weight function $x^{-k}{\mathrm{e}}^{-x}$, $x\in [0,\mathrm{\infty })$. This infinite set of polynomials of order $n\ge k$, the smallest power of $x$ being $x^k$ in each polynomial, is a complete orthogonal set with respect to this measure. These results are proven in Everitt et al. (2004), via construction of a self-adjoint Sturm–Liouville operator which generates the $L_n^{(-k)}(x)$ polynomials, self-adjointness implying both orthogonality and completeness. … ►The $y(x)={\widehat{L}}_n^{(k)}\left(x\right)$ satisfy a second order Sturm–Liouville eigenvalue problem of the type illustrated in Table 18.8.1, as satisfied by classical OP’s, but now with rational, rather than polynomial coefficients: …

17: 19.1 Special Notation

… ►The functions (19.1.1) and (19.1.2) are used in Erdélyi et al. (1953b, Chapter 13), except that $\mathrm{\Pi }({\alpha }^2,k)$ and $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ are denoted by ${\mathrm{\Pi }}_1(\nu ,k)$ and $\mathrm{\Pi }(\varphi ,\nu ,k)$, respectively, where $\nu =-{\alpha }^2$. ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha )$, $E(\alpha )$, $\mathrm{\Pi }(n\backslash \alpha )$, $F(\varphi \backslash \alpha )$, $E(\varphi \backslash \alpha )$, and $\mathrm{\Pi }(n;\varphi \backslash \alpha )$, where $\alpha =\arcsin k$ and $n$ is the ${\alpha }^2$ (not related to $k$) in (19.1.1) and (19.1.2). Also, frequently in this reference $\alpha$ is replaced by $m$ and $\mathrm{\backslash }\alpha$ by $\mathrm{|}m$, where $m=k^2$. …

18: 19.4 Derivatives and Differential Equations

… ►An analogous differential equation of third order for $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is given in Byrd and Friedman (1971, 118.03).

19: 4.2 Definitions

… ►where $k$ is the excess of the number of times the path in (4.2.1) crosses the negative real axis in the positive sense over the number of times in the negative sense. … ►

4.2.23 $$\mathrm{ph}\left(\exp z\right)=\mathrm{\Im }z+2k\pi ,$$ $k\in \mathbb{Z}$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\exp z$: exponential function, $\mathrm{\Im }$: imaginary part, $\mathbb{Z}$: set of all integers, $\mathrm{ph}$: phase, $k$: integer and $z$: complex variable

A&S Ref:

4.2.14 (modified)

Permalink:

http://dlmf.nist.gov/4.2.E23

Encodings:

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See also:

Annotations for §4.2(iii), §4.2 and Ch.4

… ►

4.2.33 $${\mathrm{e}}^z=(\exp z)\exp \left(2kz\pi \mathrm{i}\right),$$ $k\in \mathbb{Z}$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $k$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.2.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

…

20: 22.18 Mathematical Applications

… ►With $k\in [0,1]$ the mapping $z\to w=\mathrm{sn}(z,k)$ gives a conformal map of the closed rectangle $[-K,K]\times [0,K^{\prime }]$ onto the half-plane $\mathrm{\Im }w\ge 0$, with $0,\pm K,\pm K+\mathrm{i}{K}^{\prime },\mathrm{i}{K}^{\prime }$ mapping to $0,\pm 1,\pm k^{-2},\mathrm{\infty }$ respectively. … ►The special case $y^2=(1-x^2)(1-k^2{x}^2)$ is in Jacobian normal form. …