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31: 24.15 Related Sequences of Numbers

… ►

§24.15(iii) Stirling Numbers

►The Stirling numbers of the first kind $s(n,m)$, and the second kind $S(n,m)$, are as defined in §26.8(i). ►

24.15.6 $$B_n=\sum _{k=0}^n{(-1)}^k\frac{k!S(n,k)}{k+1},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $!$: factorial (as in $n!$), $k$: integer and $n$: integer

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

… ►

24.15.9 $$p\frac{B_n}{n}\equiv S(p-1+n,p-1)(mod{p}^2),$$ $1\le n\le p-2$,

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

►

24.15.10 $$\frac{2n-1}{4n}{p}^2{B}_{2n}\equiv S(p+2n,p-1)(mod{p}^3),$$ $2\le 2n\le p-3$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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32: 10.75 Tables

… ►

Achenbach (1986) tabulates $I_0\left(x\right)$, $I_1\left(x\right)$, $K_0\left(x\right)$, $K_1\left(x\right)$, $x=0(.1)8$, 19D or 19–21S.

… ►

Parnes (1972) tabulates all zeros of the principal value of $K_n\left(z\right)$, for $n=2(1)10$, 9D.

►

Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of $K_n\left(z\right)$, for $n=2(1)10$, 29S.

►

Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_n\left(z\right)$ and $K_n^{\prime }\left(z\right)$, for $n=2(1)20$, 9S.

… ►

Žurina and Karmazina (1967) tabulates ${\tilde{K}}_{\nu }\left(x\right)$ for $\nu =0.01(.01)10$, $x=0.1(.1)10.2$, 7S.

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33: 14.7 Integer Degree and Order

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14.7.2 $${𝖰}_n^0\left(x\right)={𝖰}_n\left(x\right)=\frac{1}{2}{P}_n\left(x\right)\ln \left(\frac{1+x}{1-x}\right)-W_{n-1}(x),$$

ⓘ

Symbols:

: Ferrers function of the second kind, $P_n\left(x\right)$: Legendre polynomial, $\ln z$: principal branch of logarithm function, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $n$: nonnegative integer and $W_{n-1}(x)$: quantity

A&S Ref:

8.6.19

Permalink:

http://dlmf.nist.gov/14.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.7(i), §14.7 and Ch.14

… ►

14.7.6 $$Q_n^0\left(x\right)=Q_n\left(x\right)=n!{𝑸}_n^0\left(x\right)=n!{𝑸}_n\left(x\right),$$

ⓘ

Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $!$: factorial (as in $n!$), $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $n$: nonnegative integer

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.7(i), §14.7 and Ch.14

… ►

14.7.7 $$Q_n\left(x\right)=\frac{1}{2}{P}_n\left(x\right)\ln \left(\frac{x+1}{x-1}\right)-W_{n-1}(x),$$ $n=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\ln z$: principal branch of logarithm function, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable, $n$: nonnegative integer and $W_{n-1}(x)$: quantity

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.7(i), §14.7 and Ch.14

… ►

14.7.12 $$Q_n^m\left(x\right)={\left(x^2-1\right)}^{m/2}\frac{d^m}{{dx}^m}{Q}_n\left(x\right),$$

ⓘ

Symbols:

$Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/14.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.7(ii), §14.7 and Ch.14

… ►

14.7.22 $$\sum _{n=0}^{\mathrm{\infty }}{Q}_n\left(x\right)h^n=\frac{1}{{\left(1-2xh+h^2\right)}^{1/2}}\ln \left(\frac{x-h+{\left(1-2xh+h^2\right)}^{1/2}}{{\left(x^2-1\right)}^{1/2}}\right).$$

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable, $n$: nonnegative integer and $h$: variable

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.7.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §14.7(iv), §14.7 and Ch.14

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34: 19.9 Inequalities

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$1\le E\left(k\right)\le \pi /2.$

… ►

19.9.8

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(i), §19.9 and Ch.19

►Further inequalities for $K\left(k\right)$ and $E\left(k\right)$ can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996). … ►

19.9.9 $$L(a,b)=4aE\left(k\right),$$ $k^2=1-(b^2/a^2)$, $a>b$.

ⓘ

Symbols:

$E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $k$: real or complex modulus and $L(a,b)$: perimeter

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(i), §19.9 and Ch.19

… ►Inequalities for both $F(\varphi ,k)$ and $E(\varphi ,k)$ involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). …

35: 10.1 Special Notation

… ►The main functions treated in this chapter are the Bessel functions $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$; Hankel functions $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$; modified Bessel functions $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$; spherical Bessel functions ${𝗃}_n\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗁}_n^{(1)}\left(z\right)$, ${𝗁}_n^{(2)}\left(z\right)$; modified spherical Bessel functions ${𝗂}_n^{(1)}\left(z\right)$, ${𝗂}_n^{(2)}\left(z\right)$, ${𝗄}_n\left(z\right)$; Kelvin functions ${\mathrm{ber}}_{\nu }\left(x\right)$, ${\mathrm{bei}}_{\nu }\left(x\right)$, ${\ker }_{\nu }\left(x\right)$, ${\mathrm{kei}}_{\nu }\left(x\right)$. … ►Jeffreys and Jeffreys (1956): ${\mathrm{Hs}}_{\nu }(z)$ for $H_{\nu }^{(1)}\left(z\right)$, ${\mathrm{Hi}}_{\nu }(z)$ for $H_{\nu }^{(2)}\left(z\right)$, ${\mathrm{Kh}}_{\nu }(z)$ for $(2/\pi )K_{\nu }\left(z\right)$. ►Whittaker and Watson (1927): $K_{\nu }\left(z\right)$ for $\cos \left(\nu \pi \right)K_{\nu }\left(z\right)$. ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

36: 14.1 Special Notation

… ►The main functions treated in this chapter are the Legendre functions ${𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }\left(x\right)$, $P_{\nu }\left(z\right)$, $Q_{\nu }\left(z\right)$; Ferrers functions ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P_{\nu }^{\mu }\left(z\right)$, $Q_{\nu }^{\mu }\left(z\right)$, ${𝑸}_{\nu }^{\mu }\left(z\right)$; conical functions ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${𝖰}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $P_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, $Q_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ (also known as Mehler functions). … ►Magnus et al. (1966) denotes ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$, $P_{\nu }^{\mu }\left(z\right)$, and $Q_{\nu }^{\mu }\left(z\right)$ by $P_{\nu }^{\mu }(x)$, $Q_{\nu }^{\mu }(x)$, ${𝔓}_{\nu }^{\mu }(z)$, and ${𝔔}_{\nu }^{\mu }(z)$, respectively. Hobson (1931) denotes both ${𝖯}_{\nu }^{\mu }\left(x\right)$ and $P_{\nu }^{\mu }\left(x\right)$ by $P_{\nu }^{\mu }\left(x\right)$; similarly for ${𝖰}_{\nu }^{\mu }\left(x\right)$ and $Q_{\nu }^{\mu }\left(x\right)$.

37: 10.40 Asymptotic Expansions for Large Argument

… ►

10.40.4 $$K_{\nu }^{\prime }\left(z\right)\sim -{\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.4

Referenced by:

§10.40(i), §10.40(i), §10.67(i)

Permalink:

http://dlmf.nist.gov/10.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

►Corresponding expansions for $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$, $I_{\nu }^{\prime }\left(z\right)$, and $K_{\nu }^{\prime }\left(z\right)$ for other ranges of $\mathrm{ph}z$ are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). … ►

10.40.6 $$I_{\nu }\left(z\right)K_{\nu }\left(z\right)\sim \frac{1}{2z}\left(1-\frac{1}{2}\frac{\mu -1}{{(2z)}^2}+\frac{1\cdot 3}{2\cdot 4}\frac{(\mu -1)(\mu -9)}{{(2z)}^4}-\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.7.5

Referenced by:

§10.40(i), §10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10

►

10.40.7 $$I_{\nu }^{\prime }\left(z\right)K_{\nu }^{\prime }\left(z\right)\sim -\frac{1}{2z}\left(1+\frac{1}{2}\frac{\mu -3}{{(2z)}^2}-\frac{1}{2\cdot 4}\frac{(\mu -1)(\mu -45)}{{(2z)}^4}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.7.6

Referenced by:

§10.40(i), §10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40(i), §10.40 and Ch.10

… ►

10.40.10 $$K_{\nu }\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\left(\sum _{k=0}^{\mathrm{\ell }-1}\frac{a_k(\nu )}{z^k}+R_{\mathrm{\ell }}(\nu ,z)\right),$$ $\mathrm{\ell }=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $a_k(\nu )$: polynomial coefficient and $R_{\mathrm{\ell }}(\nu ,z)$: remainder

Referenced by:

§10.40(i), §10.40(iii), §10.40(iv)

Permalink:

http://dlmf.nist.gov/10.40.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

…

38: 22.11 Fourier and Hyperbolic Series

… ►Next, with $E=E\left(k\right)$ denoting the complete elliptic integral of the second kind (§19.2(ii)) and , ►

22.11.13 $${\mathrm{sn}}^2(z,k)=\frac{1}{k^2}\left(1-\frac{E}{K}\right)-\frac{2{\pi }^2}{k^2{K}^2}\sum _{n=1}^{\mathrm{\infty }}\frac{n{q}^n}{1-q^{2n}}\cos \left(2n\zeta \right).$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§22.11

Permalink:

http://dlmf.nist.gov/22.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

… ►

22.11.14 $$k^2{\mathrm{sn}}^2(z,k)=\frac{E^{\prime }}{K^{\prime }}-{\left(\frac{\pi }{2K^{\prime }}\right)}^2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\left({\mathrm{sech}}^2\left(\frac{\pi }{2K^{\prime }}(z-2nK)\right)\right),$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $E^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the second kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{sech}z$: hyperbolic secant function, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

►where $E^{\prime }=E^{\prime }\left(k\right)$ is defined by §19.2.9. …

39: 19.11 Addition Theorems

… ►

19.11.2 $$E(\theta ,k)+E(\varphi ,k)=E(\psi ,k)+k^2\sin \theta \sin \varphi \sin \psi .$$

ⓘ

Symbols:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Permalink:

http://dlmf.nist.gov/19.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(i), §19.11 and Ch.19

… ►

19.11.8 $$E(\varphi ,k)=E\left(k\right)-E(\theta ,k)+k^2\sin \theta \sin \varphi ,$$

ⓘ

Symbols:

$E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\theta$: amplitude

Permalink:

http://dlmf.nist.gov/19.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

… ►

19.11.13 $$E(\psi ,k)=2E(\theta ,k)-k^2{\sin }^2\theta \sin \psi ,$$

ⓘ

Symbols:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Permalink:

http://dlmf.nist.gov/19.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

…

40: 12.7 Relations to Other Functions

… ►

12.7.8 $$U(-2,z)=\frac{z^{5/2}}{4\sqrt{2\pi }}\left(2K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)+3K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)-K_{\frac{5}{4}}\left(\frac{1}{4}{z}^2\right)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.11 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

►

12.7.9 $$U(-1,z)=\frac{z^{3/2}}{2\sqrt{2\pi }}\left(K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)+K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.10 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

►

12.7.10 $$U(0,z)=\sqrt{\frac{z}{2\pi }}{K}_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.9 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

►

12.7.11 $$U(1,z)=\frac{z^{3/2}}{\sqrt{2\pi }}\left(K_{\frac{3}{4}}\left(\frac{1}{4}{z}^2\right)-K_{\frac{1}{4}}\left(\frac{1}{4}{z}^2\right)\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $U(a,z)$: parabolic cylinder function and $z$: complex variable

A&S Ref:

19.15.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §12.7(iii), §12.7 and Ch.12

…