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21: 26.8 Set Partitions: Stirling Numbers

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§26.8(i) Definitions

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§26.8(ii) Generating Functions

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§26.8(iv) Recurrence Relations

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§26.8(v) Identities

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22: 22.11 Fourier and Hyperbolic Series

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22.11.3 $$\mathrm{dn}(z,k)=\frac{\pi }{2K}+\frac{2\pi }{K}\sum _{n=1}^{\mathrm{\infty }}\frac{q^n\cos \left(2n\zeta \right)}{1+q^{2n}}.$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: 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, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.3

Permalink:

http://dlmf.nist.gov/22.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.6 $$\mathrm{nd}(z,k)=\frac{\pi }{2K{k}^{\prime }}+\frac{2\pi }{K{k}^{\prime }}\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^n\cos \left(2n\zeta \right)}{1+q^{2n}}.$$

ⓘ

Symbols:

$\mathrm{nd}(z,k)$: 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, $\cos z$: cosine function, $q$: nome, $z$: complex, $k$: modulus, $k^{\prime }$: complementary modulus and $\zeta$: change of variable

A&S Ref:

16.23.6

Permalink:

http://dlmf.nist.gov/22.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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22.11.7 $$\mathrm{ns}(z,k)-\frac{\pi }{2K}\csc \zeta =\frac{2\pi }{K}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{2n+1}\sin \left((2n+1)\zeta \right)}{1-q^{2n+1}},$$

ⓘ

Symbols:

$\mathrm{ns}(z,k)$: 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, $\csc z$: cosecant function, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.10

Referenced by:

§22.11

Permalink:

http://dlmf.nist.gov/22.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

►

22.11.8 $$\mathrm{ds}(z,k)-\frac{\pi }{2K}\csc \zeta =-\frac{2\pi }{K}\sum _{n=0}^{\mathrm{\infty }}\frac{q^{2n+1}\sin \left((2n+1)\zeta \right)}{1+q^{2n+1}},$$

ⓘ

Symbols:

$\mathrm{ds}(z,k)$: 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, $\csc z$: cosecant function, $q$: nome, $\sin z$: sine function, $z$: complex, $k$: modulus and $\zeta$: change of variable

A&S Ref:

16.23.11

Permalink:

http://dlmf.nist.gov/22.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.11 and Ch.22

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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

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23: 14.6 Integer Order

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14.6.1 $${𝖯}_{\nu }^m\left(x\right)={(-1)}^m{\left(1-x^2\right)}^{m/2}\frac{d^m{𝖯}_{\nu }\left(x\right)}{{dx}^m},$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.6.6

Permalink:

http://dlmf.nist.gov/14.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(i), §14.6 and Ch.14

►

14.6.2 $${𝖰}_{\nu }^m\left(x\right)={(-1)}^m{\left(1-x^2\right)}^{m/2}\frac{d^m{𝖰}_{\nu }\left(x\right)}{{dx}^m}.$$

ⓘ

Symbols:

: Ferrers function of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.6.7

Permalink:

http://dlmf.nist.gov/14.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(i), §14.6 and Ch.14

►

14.6.3 $$P_{\nu }^m\left(x\right)={\left(x^2-1\right)}^{m/2}\frac{d^m{P}_{\nu }\left(x\right)}{{dx}^m},$$

ⓘ

Symbols:

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

A&S Ref:

8.6.6

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(i), §14.6 and Ch.14

… ►

14.6.6 $${𝖯}_{\nu }^{-m}\left(x\right)={\left(1-x^2\right)}^{-m/2}{\int }_x^1\mathrm{\dots }{\int }_x^1{𝖯}_{\nu }\left(x\right){\left(dx\right)}^m.$$

ⓘ

Symbols:

: Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.12(i), Erratum (V1.0.28) for Equation (14.6.6)

Permalink:

http://dlmf.nist.gov/14.6.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

The right-hand side has been corrected by replacing the Legendre function $P_{\nu }\left(x\right)$ with the Ferrers function ${𝖯}_{\nu }\left(x\right)$.

See also:

Annotations for §14.6(ii), §14.6 and Ch.14

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14.6.7 $$P_{\nu }^{-m}\left(x\right)={\left(x^2-1\right)}^{-m/2}{\int }_1^x\mathrm{\dots }{\int }_1^x{P}_{\nu }\left(x\right){\left(dx\right)}^m,$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$ , $\int$: integral, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.14.17

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(ii), §14.6 and Ch.14

…

24: 14.17 Integrals

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14.17.3 $$\int x{𝖯}_{\nu }^{\mu }\left(x\right){𝖰}_{\nu }^{\mu }\left(x\right)dx=\frac{1}{2\nu (\nu +1)}\left(({\mu }^2-(\nu +1)(\nu +x^2)){𝖯}_{\nu }^{\mu }\left(x\right){𝖰}_{\nu }^{\mu }\left(x\right)+(\nu +1)(\nu -\mu +1)x({𝖯}_{\nu }^{\mu }\left(x\right){𝖰}_{\nu +1}^{\mu }\left(x\right)+{𝖯}_{\nu +1}^{\mu }\left(x\right){𝖰}_{\nu }^{\mu }\left(x\right))-{(\nu -\mu +1)}^2{𝖯}_{\nu +1}^{\mu }\left(x\right){𝖰}_{\nu +1}^{\mu }\left(x\right)\right),$$ $\nu \ne 0,-1$.

ⓘ

Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(i), §14.17 and Ch.14

►

14.17.4 $$\int \frac{x}{{\left(1-x^2\right)}^{3/2}}{𝖯}_{\nu }^{\mu }\left(x\right){𝖰}_{\nu }^{\mu }\left(x\right)dx=\frac{1}{\left(1-4{\mu }^2\right){\left(1-x^2\right)}^{1/2}}\left((1-2{\mu }^2+2\nu (\nu +1)){𝖯}_{\nu }^{\mu }\left(x\right){𝖰}_{\nu }^{\mu }\left(x\right)+(2\nu +1)(\mu -\nu -1)x({𝖯}_{\nu }^{\mu }\left(x\right){𝖰}_{\nu +1}^{\mu }\left(x\right)+{𝖯}_{\nu +1}^{\mu }\left(x\right){𝖰}_{\nu }^{\mu }\left(x\right))+2{(\mu -\nu -1)}^2{𝖯}_{\nu +1}^{\mu }\left(x\right){𝖰}_{\nu +1}^{\mu }\left(x\right)\right),$$ $\mu \ne \pm \frac{1}{2}$.

ⓘ

Symbols:

: Ferrers function of the first kind, : Ferrers function of the second kind, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.17(i)

Permalink:

http://dlmf.nist.gov/14.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(i), §14.17 and Ch.14

►In (14.17.1)–(14.17.4), $𝖯$ may be replaced by $𝖰$, and in (14.17.3) and (14.17.4), $𝖰$ may be replaced by $𝖯$. … ►

14.17.15 $${\int }_{-1}^1{𝖯}_{\nu }\left(x\right){𝖰}_{\nu }\left(x\right)dx=-\frac{\sin \left(2\nu \pi \right){\psi }^{\prime }\left(\nu +1\right)}{\pi (2\nu +1)},$$ $\mathrm{\Re }\nu >0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $\sin z$: sine function, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.9

Permalink:

http://dlmf.nist.gov/14.17.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

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14.17.17 $${\int }_0^{\pi }{𝖰}_l\left(\cos \theta \right){𝖯}_m\left(\cos \theta \right){𝖯}_n\left(\cos \theta \right)\sin \theta d\theta =0,$$ $l,m,n=1,2,3,\mathrm{\dots }$, .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $\sin z$: sine function, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§14.17(iv)

Permalink:

http://dlmf.nist.gov/14.17.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

…

25: 14.11 Derivatives with Respect to Degree or Order

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14.11.1 $$\frac{\partial }{\partial \nu }{𝖯}_{\nu }^{\mu }\left(x\right)=\pi \cot \left(\nu \pi \right){𝖯}_{\nu }^{\mu }\left(x\right)-\frac{1}{\pi }{𝖠}_{\nu }^{\mu }(x),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $x$: real variable, $\mu$: general order, $\nu$: general degree and ${𝖠}_{\nu }^{\mu }(x)$

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

… ►

14.11.4 $${\frac{\partial }{\partial \mu }{𝖯}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖯}_{\nu }\left(x\right)+{𝖰}_{\nu }\left(x\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11, §14.11

Permalink:

http://dlmf.nist.gov/14.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

►

14.11.5 $${\frac{\partial }{\partial \mu }{𝖰}_{\nu }^{\mu }\left(x\right)|}_{\mu =0}=-\frac{1}{4}{\pi }^2{𝖯}_{\nu }\left(x\right)+\left(\psi \left(-\nu \right)-\pi \cot \left(\nu \pi \right)\right){𝖰}_{\nu }\left(x\right).$$

ⓘ

Symbols:

: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cot z$: cotangent function, $\psi \left(z\right)$: psi (or digamma) function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.11

Permalink:

http://dlmf.nist.gov/14.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.11 and Ch.14

►(14.11.1) holds if ${𝖯}_{\nu }^{\mu }\left(x\right)$ is replaced by $P_{\nu }^{\mu }\left(x\right)$, provided that the factor ${((1+x)/(1-x))}^{\mu /2}$ in (14.11.3) is replaced by ${((x+1)/(x-1))}^{\mu /2}$. (14.11.4) holds if ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖯}_{\nu }\left(x\right)$, and ${𝖰}_{\nu }\left(x\right)$ are replaced by $P_{\nu }^{\mu }\left(x\right)$, $P_{\nu }\left(x\right)$, and $Q_{\nu }\left(x\right)$, respectively. …

26: 29.8 Integral Equations

… ►Let $w(z)$ be any solution of (29.2.1) of period $4K$, $w_2(z)$ be a linearly independent solution, and $𝒲\left\{w,w_2\right\}$ denote their Wronskian. … ►

29.8.2 $$\mu w(z_1)w(z_2)w(z_3)={\int }_{-2K}^{2K}{𝖯}_{\nu }\left(x\right)w(z)dz,$$

ⓘ

Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $w(z)$: solution and $\mu$

Referenced by:

§29.8

Permalink:

http://dlmf.nist.gov/29.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

►where ${𝖯}_{\nu }\left(x\right)$ is the Ferrers function of the first kind (§14.3(i)), … ►

$$w(z+2K)=\sigma w(z),$$

… ►

29.8.5 $${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z_1,k^2)\frac{w_2(K)-w_2(-K)}{{d{w}_2(z)/dz|}_{z=0}}={\int }_{-K}^K{𝖯}_{\nu }\left(y\right){\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)dz,$$

ⓘ

Symbols:

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $m$: nonnegative integer, $y$: real variable, $z$: complex variable, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

…

27: 14.2 Differential Equations

… ►Standard solutions: ${𝖯}_{\nu }\left(\pm x\right)$, ${𝖰}_{\nu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}\left(\pm x\right)$, $P_{\nu }\left(\pm x\right)$, $Q_{\nu }\left(\pm x\right)$, $Q_{-\nu -1}\left(\pm x\right)$. ${𝖯}_{\nu }\left(x\right)$ and ${𝖰}_{\nu }\left(x\right)$ are real when $\nu \in ℝ$ and $x\in (-1,1)$, and $P_{\nu }\left(x\right)$ and $Q_{\nu }\left(x\right)$ are real when $\nu \in ℝ$ and $x\in (1,\mathrm{\infty })$. … ►Standard solutions: ${𝖯}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖯}_{\nu }^{-\mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}^{\mu }\left(\pm x\right)$, $P_{\nu }^{\mu }\left(\pm x\right)$, $P_{\nu }^{-\mu }\left(\pm x\right)$, ${𝑸}_{\nu }^{\mu }\left(\pm x\right)$, ${𝑸}_{-\nu -1}^{\mu }\left(\pm x\right)$. … ►Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations ${𝖯}_{\nu }^0\left(x\right)={𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }^0\left(x\right)={𝖰}_{\nu }\left(x\right)$, $P_{\nu }^0\left(x\right)=P_{\nu }\left(x\right)$, $Q_{\nu }^0\left(x\right)=Q_{\nu }\left(x\right)$, ${𝑸}_{\nu }^0\left(x\right)={𝑸}_{\nu }\left(x\right)=Q_{\nu }\left(x\right)/\mathrm{\Gamma }\left(\nu +1\right)$. ► ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$, and ${𝖰}_{\nu }^{\mu }\left(x\right)$ are real when $\nu$, $\mu$, and $\tau \in ℝ$, and $x\in (-1,1)$; $P_{\nu }^{\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ are real when $\nu$ and $\mu \in ℝ$, and $x\in (1,\mathrm{\infty })$. …

28: 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). … ►Among other notations commonly used in the literature Erdélyi et al. (1953a) and Olver (1997b) denote ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ by ${\mathrm{P}}_{\nu }^{\mu }(x)$ and ${\mathrm{Q}}_{\nu }^{\mu }(x)$, respectively. 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 $Q_{\nu }^{\mu }\left(x\right)$.

29: 14.7 Integer Degree and Order

… ►

14.7.1 $${𝖯}_n^0\left(x\right)={𝖯}_n\left(x\right)=P_n^0\left(x\right)=P_n\left(x\right),$$ $x\in ℝ$,

ⓘ

Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $P_n\left(x\right)$: Legendre polynomial, $\in$: element of, $ℝ$: real line, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/14.7.E1

Encodings:

pMML, png, TeX

See also:

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

… ►

14.7.8 $${𝖯}_n^m\left(x\right)={(-1)}^m{\left(1-x^2\right)}^{m/2}\frac{d^m}{{dx}^m}{𝖯}_n\left(x\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/14.7.E8

Encodings:

pMML, png, TeX

See also:

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

… ►When $m$ is even and $m\le n$, ${𝖯}_n^m\left(x\right)$ and $P_n^m\left(x\right)$ are polynomials of degree $n$. … ►

14.7.16 $${𝖯}_n^m\left(x\right)=P_n^m\left(x\right)=0,$$ $m>n$.

ⓘ

Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§14.18(ii), §14.21(iii)

Permalink:

http://dlmf.nist.gov/14.7.E16

Encodings:

pMML, png, TeX

See also:

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

… ►

14.7.17 $${𝖯}_n^m\left(-x\right)={(-1)}^{n-m}{𝖯}_n^m\left(x\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $x$: real variable, $m$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§14.18(iii), §14.30(ii)

Permalink:

http://dlmf.nist.gov/14.7.E17

Encodings:

pMML, png, TeX

See also:

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

…

30: 10.44 Sums

… ►If $𝒵=I$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary. … ►

$I_{\nu }\left(z\right)={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{z^k}{k!}}{J}_{\nu +k}\left(z\right),$

… ►The restriction is unnecessary when $𝒵=I$ and $\nu$ is an integer. … ►

10.44.5 $$K_0\left(z\right)=-\left(\ln \left(\frac{1}{2}z\right)+\gamma \right)I_0\left(z\right)+2\sum _{k=1}^{\mathrm{\infty }}\frac{I_{2k}\left(z\right)}{k},$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.54

Permalink:

http://dlmf.nist.gov/10.44.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.44(iii), §10.44 and Ch.10

►

10.44.6 $$K_n\left(z\right)=\frac{n!{(\frac{1}{2}z)}^{-n}}{2}\sum _{k=0}^{n-1}{(-1)}^k\frac{{(\frac{1}{2}z)}^k{I}_k\left(z\right)}{k!(n-k)}+{(-1)}^{n-1}\left(\ln \left(\frac{1}{2}z\right)-\psi \left(n+1\right)\right)I_n\left(z\right)+{(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{(n+2k)I_{n+2k}\left(z\right)}{k(n+k)},$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.53

Permalink:

http://dlmf.nist.gov/10.44.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.44(iii), §10.44 and Ch.10

…