DLMF: Notations S

Notations S

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$\mathop{\mathfrak{S}_{{n}}\/}\nolimits$: set of permutations of $\{ 1,2,\ldots,n\}$ ; §26.13
$\mathscr{S}^{{(k)}}_{n}=\mathop{S\/}\nolimits\!\left(n,k\right)$: notation used by Fort (1948); ¶ ‣ §26.1
(with $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind)
$S_{n}^{{(k)}}=\mathop{s\/}\nolimits\!\left(n,k\right)$: notation used by Abramowitz and Stegun (1964, Chapter 24), Fort (1948); ¶ ‣ §26.1
(with $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind)
$S_{1}(n-1,n-k)=\mathop{s\/}\nolimits\!\left(n,k\right)/(-1)^{{n-k}}$: notation used by Carlitz (1960), Gould (1960); ¶ ‣ §26.1
(with $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind)
$\mathfrak{S}_{n}^{k}=\mathop{S\/}\nolimits\!\left(n,k\right)$: notation used by Jordan (1939); ¶ ‣ §26.1
(with $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind)
$S_{n}^{k}=\mathop{s\/}\nolimits\!\left(n,k\right)$: notation used by (Jordan, 1939), Moser and Wyman (1958a); ¶ ‣ §26.1
(with $\mathop{s\/}\nolimits\!\left(n,k\right)$ : Stirling number of the first kind)
$\mathop{S\/}\nolimits\!\left(z\right)$: Fresnel integral; (7.2.8)
$S_{1}(z)=\mathop{S\/}\nolimits\!\left(\sqrt{2/\pi}z\right)$: alternative notation for the Fresnel integral; §7.1
(with $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral)
$S_{2}(z)=\mathop{S\/}\nolimits\!\left(\sqrt{2z/\pi}\right)$: alternative notation for the Fresnel integral; §7.1
(with $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral)
$\mathop{S_{{n}}\/}\nolimits\!\left(x\right)$: dilated Chebyshev polynomial; (18.1.3)
$\mathop{S_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$: Lommel function; (11.9.5)
$\mathop{s_{{{\mu},{\nu}}}\/}\nolimits\!\left(z\right)$: Lommel function; (11.9.3)
$\mathop{S^{{m(j)}}_{{n}}\/}\nolimits\!\left(z,\gamma\right)$: radial spheroidal wave function; (30.11.3)
$\mathop{\mathcal{S}\/}\nolimits\left(f;s\right)$: Stieltjes transform; (1.14.47)
$\mathop{S\/}\nolimits\!\left(n,k\right)$: Stirling number of the second kind; §26.8(i)
$\mathop{s\/}\nolimits\!\left(n,k\right)$: Stirling number of the first kind; §26.8(i)
$S_{2}(k,n-k)=\mathop{S\/}\nolimits\!\left(n,k\right)$: notation used by Carlitz (1960), Gould (1960); ¶ ‣ §26.1
(with $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind)
$\mathop{S_{{n}}\/}\nolimits\!\left(x;q\right)$: Stieltjes–Wigert polynomial; (18.27.18)
$S^{{(1)}}_{{mn}}(\gamma,x)\propto\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$: alternative notation for the spheroidal wave function of the first kind; ¶ ‣ §30.1
(with $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind)
$S^{{(2)}}_{{mn}}(\gamma,x)\propto\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$: alternative notation for the spheroidal wave function of the second kind; ¶ ‣ §30.1
(with $\mathop{\mathsf{Qs}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the second kind)
$S_{n}^{m}(z,\xi)$: Ince polynomials; §28.31(ii)
$\mathop{s\/}\nolimits\!\left(\epsilon,\ell;r\right)$: regular Coulomb function; (33.14.9)
$S(k,h)(x)$: Sinc function; §3.3(vi)
$\mathop{S_{{n}}\/}\nolimits\!\left(x;a,b,c\right)$: continuous dual Hahn polynomial; Table 18.25.1
$\mathop{\mathrm{sc}\/}\nolimits\left(z,k\right)$: Jacobian elliptic function; (22.2.9)
$\mathop{\mathit{scdE}^{{m}}_{{2n+3}}\/}\nolimits\!\left(z,k^{2}\right)$: Lamé polynomial; (29.12.8)
$\mathop{\mathit{scE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$: Lamé polynomial; (29.12.5)
$\mathop{\mathrm{sd}\/}\nolimits\left(z,k\right)$: Jacobian elliptic function; (22.2.7)
$\mathop{\mathit{sdE}^{{m}}_{{2n+2}}\/}\nolimits\!\left(z,k^{2}\right)$: Lamé polynomial; (29.12.6)
$\mathop{\mathrm{Se}_{{\nu}}\/}\nolimits\!\left(z,q\right)$: modified Mathieu function; (28.20.4)

$\mathrm{Se}_{{n}}(s,z)=\dfrac{\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)}{\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(0,q\right)}$: notation used by National Bureau of Standards (1967); ¶ ‣ §28.1
(with $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function)
$\mathrm{Se}_{{n}}(c,z)=\dfrac{\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)}{\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(0,q\right)}$: notation used by Stratton et al. (1941); ¶ ‣ §28.1
(with $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function)
$\mathop{\mathrm{se}_{{\nu}}\/}\nolimits\!\left(z,q\right)$: Mathieu function of noninteger order; (28.12.13)
$\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$: Mathieu function; §28.2(vi)
$\mathop{\mathit{sE}^{{m}}_{{2n+1}}\/}\nolimits\!\left(z,k^{2}\right)$: Lamé polynomial; (29.12.2)
$\mathop{\sec\/}\nolimits z$: secant function; (4.14.6)
$\mathop{\mathrm{sech}\/}\nolimits z$: hyperbolic secant function; (4.28.6)
$\mathrm{seh}_{n}(z,q)=\mathop{\mathrm{Se}_{{n}}\/}\nolimits\!\left(z,q\right)$: notation used by Campbell (1955); ¶ ‣ §28.1
(with $\mathop{\mathrm{Se}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function)
$\mathop{\mathrm{Shi}\/}\nolimits\!\left(z\right)$: hyperbolic sine integral; (6.2.15)
$\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$: sine integral; (6.2.9)
$\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)$: sine integral; (6.2.10)
$\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$: generalized sine integral; (8.21.2)
$\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$: generalized sine integral; (8.21.1)
$\sigma _{n}^{k}=\mathop{S\/}\nolimits\!\left(n,k\right)$: notation used by Moser and Wyman (1958b); ¶ ‣ §26.1
(with $\mathop{S\/}\nolimits\!\left(n,k\right)$ : Stirling number of the second kind)
$\mathop{\sigma _{{n}}\/}\nolimits\!\left(\nu\right)$: Rayleigh function; (10.21.55)
$\mathop{{\sigma _{{\ell}}}\/}\nolimits\!\left(\eta\right)$: Coulomb phase shift; (33.2.10)
$\mathop{\sigma _{{\alpha}}\/}\nolimits\!\left(n\right)$: sum of powers of divisors of $n$ ; (27.2.10)
$\mathop{\sigma\/}\nolimits\!\left(z\right)$ (= $\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass sigma function; (23.2.6)
$\mathop{\sigma\/}\nolimits\!\left(z|\mathbb{L}\right)$: Weierstrass sigma function; §23.1
$\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$: Weierstrass sigma function; §23.3(i)
$\mathop{\mathrm{sign}\/}\nolimits x$: sign of $x$ ; Common Notations and Definitions
$\mathop{\sin\/}\nolimits z$: sine function; (4.14.1)
$\mathop{\mathrm{Sin}_{{q}}\/}\nolimits\!\left(x\right)$: $q$ -sine function; (17.3.4)
$\mathop{\mathrm{sin}_{{q}}\/}\nolimits\!\left(x\right)$: $q$ -sine function; (17.3.3)
$\mathop{\sinh\/}\nolimits z$: hyperbolic sine function; (4.28.1)
$\mathrm{sn}(z\mathpunct{|}m)=\mathop{\mathrm{sn}\/}\nolimits\left(z,\sqrt{m}\right)$: alternative notation; §22.1
(with $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function)
$\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$: Jacobian elliptic function; (22.2.4)
$\mathrm{So}_{{n}}(s,z)=\dfrac{\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)}{{\mathop{\mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)}$: notation used by National Bureau of Standards (1967); ¶ ‣ §28.1
(with $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function)
$\mathrm{So}_{{n}}(c,z)=\dfrac{\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)}{{\mathop{\mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(0,q\right)}$: notation used by Stratton et al. (1941); ¶ ‣ §28.1
(with $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function)
$\sup$: least upper bound (supremum); Common Notations and Definitions