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21: 14.10 Recurrence Relations and Derivatives

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14.10.1 $${𝖯}_{\nu }^{\mu +2}\left(x\right)+2(\mu +1)x{\left(1-x^2\right)}^{-1/2}{𝖯}_{\nu }^{\mu +1}\left(x\right)+(\nu -\mu )(\nu +\mu +1){𝖯}_{\nu }^{\mu }\left(x\right)=0,$$

ⓘ

Symbols:

: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.10

Permalink:

http://dlmf.nist.gov/14.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

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14.10.2 $${\left(1-x^2\right)}^{1/2}{𝖯}_{\nu }^{\mu +1}\left(x\right)-(\nu -\mu +1){𝖯}_{\nu +1}^{\mu }\left(x\right)+(\nu +\mu +1)x{𝖯}_{\nu }^{\mu }\left(x\right)=0,$$

ⓘ

Symbols:

: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

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14.10.3 $$(\nu -\mu +2){𝖯}_{\nu +2}^{\mu }\left(x\right)-(2\nu +3)x{𝖯}_{\nu +1}^{\mu }\left(x\right)+(\nu +\mu +1){𝖯}_{\nu }^{\mu }\left(x\right)=0,$$

ⓘ

Symbols:

: Ferrers function of the first kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.5.3 (modified)

Referenced by:

§14.10, §14.14, §14.21(iii)

Permalink:

http://dlmf.nist.gov/14.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

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14.10.4 $$\left(1-x^2\right)\frac{d{𝖯}_{\nu }^{\mu }\left(x\right)}{dx}=(\mu -\nu -1){𝖯}_{\nu +1}^{\mu }\left(x\right)+(\nu +1)x{𝖯}_{\nu }^{\mu }\left(x\right),$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.5(i)

Permalink:

http://dlmf.nist.gov/14.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

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14.10.5 $$\left(1-x^2\right)\frac{d{𝖯}_{\nu }^{\mu }\left(x\right)}{dx}=(\nu +\mu ){𝖯}_{\nu -1}^{\mu }\left(x\right)-\nu x{𝖯}_{\nu }^{\mu }\left(x\right).$$

ⓘ

Symbols:

: Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.5.4 (modified)

Referenced by:

§14.10

Permalink:

http://dlmf.nist.gov/14.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.10 and Ch.14

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22: 30.4 Functions of the First Kind

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30.4.1 $${\int }_{-1}^1{\left({\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right)}^{2}dx=\frac{2}{2n+1}\frac{(n+m)!}{(n-m)!},$$

ⓘ

Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

A&S Ref:

21.7.11

Referenced by:

§30.4(iii)

Permalink:

http://dlmf.nist.gov/30.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(i), §30.4 and Ch.30

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30.4.2 $${\mathrm{𝖯𝗌}}_n^m(x,0)={𝖯}_n^m\left(x\right);$$

ⓘ

Symbols:

: Ferrers function of the first kind, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer and $n\ge m$: integer degree

Referenced by:

§30.4(iv)

Permalink:

http://dlmf.nist.gov/30.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(i), §30.4 and Ch.30

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30.4.3 $${\mathrm{𝖯𝗌}}_n^m(-x,{\gamma }^2)={(-1)}^{n-m}{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2).$$

ⓘ

Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(ii), §30.4 and Ch.30

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30.4.4 $${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)={(1-x^2)}^{\frac{1}{2}m}\sum _{k=0}^{\mathrm{\infty }}{g}_k{x}^k,$$ $-1\le x\le 1$,

ⓘ

Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

A&S Ref:

21.7.18

Permalink:

http://dlmf.nist.gov/30.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iii), §30.4 and Ch.30

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30.4.7 $$f(x)=\sum _{n=m}^{\mathrm{\infty }}{c}_n{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

ⓘ

Symbols:

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $f(x)$: function and $c_n$: coefficient

Referenced by:

§30.4(iv)

Permalink:

http://dlmf.nist.gov/30.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.4(iv), §30.4 and Ch.30

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23: 14.9 Connection Formulas

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14.9.3 $${𝖯}_{\nu }^{-m}\left(x\right)={(-1)}^m\frac{\mathrm{\Gamma }\left(\nu -m+1\right)}{\mathrm{\Gamma }\left(\nu +m+1\right)}{𝖯}_{\nu }^m\left(x\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.17(iii), §14.18(ii), §14.3(i), §14.6(ii), §14.9(i)

Permalink:

http://dlmf.nist.gov/14.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(i), §14.9 and Ch.14

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14.9.4 $${𝖰}_{\nu }^{-m}\left(x\right)={(-1)}^m\frac{\mathrm{\Gamma }\left(\nu -m+1\right)}{\mathrm{\Gamma }\left(\nu +m+1\right)}{𝖰}_{\nu }^m\left(x\right),$$ $\nu \ne m-1,m-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.6(ii), §14.9(i)

Permalink:

http://dlmf.nist.gov/14.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(i), §14.9 and Ch.14

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14.9.13 $$P_{\nu }^{-m}\left(x\right)=\frac{\mathrm{\Gamma }\left(\nu -m+1\right)}{\mathrm{\Gamma }\left(\nu +m+1\right)}{P}_{\nu }^m\left(x\right),$$ $\nu \ne m-1,m-2,\mathrm{\dots }$.

ⓘ

Symbols:

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

Referenced by:

§14.19(v), §14.6(ii)

Permalink:

http://dlmf.nist.gov/14.9.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iii), §14.9 and Ch.14

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14.9.14 $${𝑸}_{\nu }^{-\mu }\left(x\right)={𝑸}_{\nu }^{\mu }\left(x\right),$$

ⓘ

Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.2.6 (modified)

Referenced by:

§14.23, §14.9(iv)

Permalink:

http://dlmf.nist.gov/14.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iii), §14.9 and Ch.14

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14.9.16 $${𝑸}_{\nu }^{\mu }\left(x\right)={\left(\frac{1}{2}\pi \right)}^{1/2}{\left(x^2-1\right)}^{-1/4}{P}_{-\mu -(1/2)}^{-\nu -(1/2)}\left(x{\left(x^2-1\right)}^{-1/2}\right).$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.2.7 (modified)

Referenced by:

§14.19(v), §14.21(iii), §14.9(iv)

Permalink:

http://dlmf.nist.gov/14.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iv), §14.9 and Ch.14

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24: 1.1 Special Notation

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$x,y$	real variables.
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$\mathrm{deg}$	degree.
…

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25: 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

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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

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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

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14.6.4 $$Q_{\nu }^m\left(x\right)={\left(x^2-1\right)}^{m/2}\frac{d^m{Q}_{\nu }\left(x\right)}{{dx}^m},$$

ⓘ

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 $\nu$: general degree

A&S Ref:

8.6.7

Referenced by:

§14.6(i)

Permalink:

http://dlmf.nist.gov/14.6.E4

Encodings:

pMML, png, TeX

See also:

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

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14.6.5 $${\left(\nu +1\right)}_m{𝑸}_{\nu }^m\left(x\right)={(-1)}^m{\left(x^2-1\right)}^{m/2}\frac{d^m{𝑸}_{\nu }\left(x\right)}{{dx}^m}.$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.21(iii), §14.6(i)

Permalink:

http://dlmf.nist.gov/14.6.E5

Encodings:

pMML, png, TeX

See also:

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

…

26: Bille C. Carlson

… ►After the war he returned to Harvard and completed Bachelor’s and Master’s degrees in physics and mathematics. He then went to Oxford as a Rhodes Scholar and completed a doctoral degree in physics. …

27: 14.4 Graphics

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28: 14.29 Generalizations

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14.29.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }_1^2}{2(1-z)}-\frac{{\mu }_2^2}{2(1+z)}\right)w=0$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.29

Permalink:

http://dlmf.nist.gov/14.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.29 and Ch.14

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29: 18.31 Bernstein–Szegő Polynomials

… ►Let $\rho (x)$ be a polynomial of degree $\mathrm{\ell }$ and positive when $-1\le x\le 1$. …

30: Daniel W. Lozier

… ►Lozier received a degree in mathematics from Oregon State University in 1962 and his Ph. …