Notations Notations R Notations T

Notations S

${𝔖}_n$

set of permutations of $\{1,2,\mathrm{\dots },n\}$; §26.13

$S_n^{(k)}=s(n,k)$

notation used by Abramowitz and Stegun (1964, Chapter 24), Fort (1948); §26.1
(with $s(n,k)$: Stirling number of the first kind)

${𝒮}_n^{(k)}=S(n,k)$

notation used by Fort (1948); §26.1
(with $S(n,k)$: Stirling number of the second kind)

${𝔖}_n^k=S(n,k)$

notation used by Jordan (1939); §26.1
(with $S(n,k)$: Stirling number of the second kind)

$S\left(z\right)$

Fresnel integral; (7.2.8)

$S_1(z)=S\left(\sqrt{2/\pi }z\right)$

alternative notation for the Fresnel integral; §7.1
(with $S\left(z\right)$: Fresnel integral and $\pi$: the ratio of the circumference of a circle to its diameter)

$S_2(z)=S\left(\sqrt{2z/\pi }\right)$

alternative notation for the Fresnel integral; §7.1
(with $S\left(z\right)$: Fresnel integral and $\pi$: the ratio of the circumference of a circle to its diameter)

$S_{\mu ,\nu }\left(z\right)$

Lommel function; (11.9.5)

$s_{\mu ,\nu }\left(z\right)$

Lommel function; (11.9.3)

$S_n\left(x\right)$

dilated Chebyshev polynomial; (18.1.3)

$𝒮\left(f\right)\left(s\right)$

Stieltjes transform; (1.14.47)

$s(n,k)$

Stirling number of the first kind; §26.8(i)

$S(n,k)$

Stirling number of the second kind; §26.8(i)

$S_1(n-1,n-k)=s(n,k)/{(-1)}^{n-k}$

notation used by Carlitz (1960), Gould (1960); §26.1
(with $s(n,k)$: Stirling number of the first kind)

$S_2(k,n-k)=S(n,k)$

notation used by Carlitz (1960), Gould (1960); §26.1
(with $S(n,k)$: Stirling number of the second kind)

$S_n^{m(j)}(z,\gamma )$

radial spheroidal wave function; (30.11.3)

$S_{mn}^{(1)}(\gamma ,x)\propto {\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$

alternative notation for the spheroidal wave function of the first kind; §30.1
(with ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind)

$S_{mn}^{(2)}(\gamma ,x)\propto {\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$

alternative notation for the spheroidal wave function of the second kind; §30.1
(with ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind)

$S_n^m(z,\xi )$

Ince polynomials; §28.31(ii)

$S_n(x;q)$

Stieltjes–Wigert polynomial; (18.27.18)

$s(ϵ,\mathrm{\ell };r)$

regular Coulomb function; (33.14.9)

$S(k,h)(x)$

Sinc function; §3.3(vi)

$S_n(x;a,b,c)$

continuous dual Hahn polynomial; Table 18.25.1

$\mathrm{sc}(z,k)$

Jacobian elliptic function; (22.2.9)

${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$

Lamé polynomial; (29.12.8)

${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$

Lamé polynomial; (29.12.5)

$\mathrm{sd}(z,k)$

Jacobian elliptic function; (22.2.7)

${\mathrm{𝑠𝑑𝐸}}_{2n+2}^m(z,k^2)$

Lamé polynomial; (29.12.6)

${\mathrm{Se}}_n(c,z)={\displaystyle \frac{{\mathrm{ce}}_n(z,q)}{{\mathrm{ce}}_n(0,q)}}$

notation used by Stratton et al. (1941); §28.1
(with ${\mathrm{ce}}_n(z,q)$: Mathieu function)

${\mathrm{Se}}_n(s,z)={\displaystyle \frac{{\mathrm{ce}}_n(z,q)}{{\mathrm{ce}}_n(0,q)}}$

notation used by National Bureau of Standards (1967); §28.1
(with ${\mathrm{ce}}_n(z,q)$: Mathieu function)

${\mathrm{se}}_n(z,q)$

Mathieu function; §28.2(vi)

${\mathrm{se}}_{\nu }(z,q)$

Mathieu function of noninteger order; (28.12.13)

${\mathrm{Se}}_{\nu }(z,q)$

modified Mathieu function; (28.20.4)

${\mathrm{𝑠𝐸}}_{2n+1}^m(z,k^2)$

Lamé polynomial; (29.12.2)

$\sec z$

secant function; (4.14.6)

$\mathrm{sech}z$

hyperbolic secant function; (4.28.6)

${\mathrm{seh}}_n(z,q)={\mathrm{Se}}_n(z,q)$

notation used by Campbell (1955); §28.1
(with ${\mathrm{Se}}_{\nu }(z,q)$: modified Mathieu function)

$\mathrm{Shi}\left(z\right)$

hyperbolic sine integral; (6.2.15)

$\mathrm{Si}\left(z\right)$

sine integral; (6.2.9)

$\mathrm{si}\left(z\right)$

sine integral; (6.2.10)

$\mathrm{Si}(a,z)$

generalized sine integral; (8.21.2)

$\mathrm{si}(a,z)$

generalized sine integral; (8.21.1)

${\sigma }_n^k=S(n,k)$

notation used by Moser and Wyman (1958b); §26.1
(with $S(n,k)$: Stirling number of the second kind)

${\sigma }_{\alpha }\left(n\right)$

sum of powers of divisors of a number; (27.2.10)

${\sigma }_{\mathrm{\ell }}\left(\eta \right)$

Coulomb phase shift; (33.2.10)

${\sigma }_n\left(\nu \right)$

Rayleigh function; (10.21.55)

$\sigma \left(z\right)$ (= $\sigma \left(z|𝕃\right)$ = $\sigma (z;g_2,g_3)$)

Weierstrass sigma function; (23.2.6)

$\sigma (z;g_2,g_3)$

Weierstrass sigma function; §23.3(i)

$\mathrm{sign}x$

sign of; Common Notations and Definitions

$\sin z$

sine function; (4.14.1)

${\mathrm{Sin}}_q\left(x\right)$

$q$-sine function; (17.3.4)

${\sin }_q\left(x\right)$

$q$-sine function; (17.3.3)

$\sinh z$

hyperbolic sine function; (4.28.1)

$\mathrm{sn}(z,k)$

Jacobian elliptic function; (22.2.4)

$\mathrm{sn}(z|m)=\mathrm{sn}(z,\sqrt{m})$

alternative notation; §22.1
(with $\mathrm{sn}(z,k)$: Jacobian elliptic function)

${\mathrm{So}}_n(c,z)={\displaystyle \frac{{\mathrm{se}}_n(z,q)}{{\mathrm{se}}_n^{\prime }(0,q)}}$

notation used by Stratton et al. (1941); §28.1
(with ${\mathrm{se}}_n(z,q)$: Mathieu function)

${\mathrm{So}}_n(s,z)={\displaystyle \frac{{\mathrm{se}}_n(z,q)}{{\mathrm{se}}_n^{\prime }(0,q)}}$

notation used by National Bureau of Standards (1967); §28.1
(with ${\mathrm{se}}_n(z,q)$: Mathieu function)

$sup$

least upper bound (supremum); Common Notations and Definitions

Notations R Notations T

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