Notations S

$\mathfrak{S}_{\NVar{n}}$: set of permutations of $\{1,2,\ldots,n\}$ ; §26.13
$S_{\NVar{n}}^{(\NVar{k})}=s\left(n,k\right)$: notation used by Abramowitz and Stegun (1964, Chapter 24), Fort (1948); §26.1
(with $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind)
$\mathscr{S}^{(\NVar{k})}_{\NVar{n}}=S\left(n,k\right)$: notation used by Fort (1948); §26.1
(with $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind)
$\mathfrak{S}_{\NVar{n}}^{\NVar{k}}=S\left(n,k\right)$: notation used by Jordan (1939); §26.1
(with $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind)
$S\left(\NVar{z}\right)$: Fresnel integral; (7.2.8)
$S_{1}(\NVar{z})=S\left(\sqrt{2/\pi}z\right)$: alternative notation for the Fresnel integral; §7.1
(with $S\left(\NVar{z}\right)$ : Fresnel integral and $\pi$ : the ratio of the circumference of a circle to its diameter)
$S_{2}(\NVar{z})=S\left(\sqrt{2z/\pi}\right)$: alternative notation for the Fresnel integral; §7.1
(with $S\left(\NVar{z}\right)$ : Fresnel integral and $\pi$ : the ratio of the circumference of a circle to its diameter)
$S_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$: Lommel function; (11.9.5)
$s_{{\NVar{\mu}},{\NVar{\nu}}}\left(\NVar{z}\right)$: Lommel function; (11.9.3)
$S_{\NVar{n}}\left(\NVar{x}\right)$: dilated Chebyshev polynomial; (18.1.3)
$\mathcal{S}\left(\NVar{f}\right)\left(\NVar{s}\right)$: Stieltjes transform; (1.14.47)
$s\left(\NVar{n},\NVar{k}\right)$: Stirling number of the first kind; §26.8(i)
$S\left(\NVar{n},\NVar{k}\right)$: Stirling number of the second kind; §26.8(i)
$S_{1}(\NVar{n-1},\NVar{n-k})=s\left(n,k\right)/(-1)^{n-k}$: notation used by Carlitz (1960), Gould (1960); §26.1
(with $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind)
$S_{2}(\NVar{k},\NVar{n-k})=S\left(n,k\right)$: notation used by Carlitz (1960), Gould (1960); §26.1
(with $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind)
$S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma}\right)$: radial spheroidal wave function; (30.11.3)
$S^{(1)}_{\NVar{mn}}(\NVar{\gamma},\NVar{x})\propto\mathsf{Ps}^{m}_{n}\left(x,% \gamma^{2}\right)$: alternative notation for the spheroidal wave function of the first kind; §30.1
(with $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind)
$S^{(2)}_{\NVar{mn}}(\NVar{\gamma},\NVar{x})\propto\mathsf{Qs}^{m}_{n}\left(x,% \gamma^{2}\right)$: alternative notation for the spheroidal wave function of the second kind; §30.1
(with $\mathsf{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the second kind)
$S_{\NVar{n}}^{\NVar{m}}(\NVar{z},\NVar{\xi})$: Ince polynomials; §28.31(ii)
$S_{\NVar{n}}\left(\NVar{x};\NVar{q}\right)$: Stieltjes–Wigert polynomial; (18.27.18)
$s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$: regular Coulomb function; (33.14.9)
$S(\NVar{k},\NVar{h})(\NVar{x})$: Sinc function; §3.3(vi)
$S_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$: continuous dual Hahn polynomial; Table 18.25.1
$\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function; (22.2.9)
$\mathit{scdE}^{\NVar{m}}_{2\NVar{n}+3}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial; (29.12.8)
$\mathit{scE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial; (29.12.5)
$\operatorname{sd}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function; (22.2.7)
$\mathit{sdE}^{\NVar{m}}_{2\NVar{n}+2}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial; (29.12.6)
$\mathrm{Se}_{\NVar{n}}(\NVar{c},\NVar{z})=\dfrac{\operatorname{ce}_{n}\left(z,% q\right)}{\operatorname{ce}_{n}\left(0,q\right)}$: notation used by Stratton et al. (1941); §28.1
(with $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function)

$\mathrm{Se}_{\NVar{n}}(\NVar{s},\NVar{z})=\dfrac{\operatorname{ce}_{n}\left(z,% q\right)}{\operatorname{ce}_{n}\left(0,q\right)}$: notation used by National Bureau of Standards (1967); §28.1
(with $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function)
$\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function; §28.2(vi)
$\operatorname{se}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function of noninteger order; (28.12.13)
$\operatorname{Se}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function; (28.20.4)
$\mathit{sE}^{\NVar{m}}_{2\NVar{n}+1}\left(\NVar{z},\NVar{k^{2}}\right)$: Lamé polynomial; (29.12.2)
$\sec\NVar{z}$: secant function; (4.14.6)
$\operatorname{sech}\NVar{z}$: hyperbolic secant function; (4.28.6)
$\mathrm{seh}_{\NVar{n}}(\NVar{z},\NVar{q})=\operatorname{Se}_{n}\left(z,q\right)$: notation used by Campbell (1955); §28.1
(with $\operatorname{Se}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function)
$\operatorname{Shi}\left(\NVar{z}\right)$: hyperbolic sine integral; (6.2.15)
$\operatorname{Si}\left(\NVar{z}\right)$: sine integral; (6.2.9)
$\operatorname{si}\left(\NVar{z}\right)$: sine integral; (6.2.10)
$\operatorname{Si}\left(\NVar{a},\NVar{z}\right)$: generalized sine integral; (8.21.2)
$\operatorname{si}\left(\NVar{a},\NVar{z}\right)$: generalized sine integral; (8.21.1)
$\sigma_{\NVar{n}}^{\NVar{k}}=S\left(n,k\right)$: notation used by Moser and Wyman (1958b); §26.1
(with $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind)
$\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$: sum of powers of divisors of a number; (27.2.10)
${\sigma_{\NVar{\ell}}}\left(\NVar{\eta}\right)$: Coulomb phase shift; (33.2.10)
$\sigma_{\NVar{n}}\left(\NVar{\nu}\right)$: Rayleigh function; (10.21.55)
$\sigma\left(\NVar{z}\right)$ (= $\sigma\left(z|\mathbb{L}\right)$ = $\sigma\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function; (23.2.6)
$\sigma\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$: Weierstrass sigma function; §23.3(i)
$\operatorname{sign}\NVar{x}$: sign of; Common Notations and Definitions
$\sin\NVar{z}$: sine function; (4.14.1)
$\operatorname{Sin}_{\NVar{q}}\left(\NVar{x}\right)$: $q$ -sine function; (17.3.4)
$\operatorname{sin}_{\NVar{q}}\left(\NVar{x}\right)$: $q$ -sine function; (17.3.3)
$\sinh\NVar{z}$: hyperbolic sine function; (4.28.1)
$\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function; (22.2.4)
$\mathrm{sn}(\NVar{z}\mathpunct{|}\NVar{m})=\operatorname{sn}\left(z,\sqrt{m}\right)$: alternative notation; §22.1
(with $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function)
$\mathrm{So}_{\NVar{n}}(\NVar{c},\NVar{z})=\dfrac{\operatorname{se}_{n}\left(z,% q\right)}{\operatorname{se}_{n}'\left(0,q\right)}$: notation used by Stratton et al. (1941); §28.1
(with $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function)
$\mathrm{So}_{\NVar{n}}(\NVar{s},\NVar{z})=\dfrac{\operatorname{se}_{n}\left(z,% q\right)}{\operatorname{se}_{n}'\left(0,q\right)}$: notation used by National Bureau of Standards (1967); §28.1
(with $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function)
$\sup$: least upper bound (supremum); Common Notations and Definitions

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