ultraspherical polynomials

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11: 18.14 Inequalities

… ►

Ultraspherical

►

18.14.4 $$|C_n^{(\lambda )}\left(x\right)|\le C_n^{(\lambda )}\left(1\right)=\frac{{\left(2\lambda \right)}_n}{n!},$$ $-1\le x\le 1$, $\lambda >0$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.14.2

Referenced by:

§18.14(i)

Permalink:

http://dlmf.nist.gov/18.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

►

18.14.5 $$|C_{2m}^{(\lambda )}\left(x\right)|\le |C_{2m}^{(\lambda )}\left(0\right)|=\left|\frac{{\left(\lambda \right)}_m}{m!}\right|,$$ $-1\le x\le 1$, ,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $m$: nonnegative integer and $x$: real variable

A&S Ref:

22.14.2

Referenced by:

§18.14(i)

Permalink:

http://dlmf.nist.gov/18.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

►

18.14.6 $-1\le x\le 1$, .

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $m$: nonnegative integer and $x$: real variable

Referenced by:

§18.14(i)

Permalink:

http://dlmf.nist.gov/18.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

►

18.14.7 $-1\le x\le 1$, .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.14(i)

Permalink:

http://dlmf.nist.gov/18.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.14(i), §18.14(i), §18.14 and Ch.18

…

12: 18.5 Explicit Representations

… ►

Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).

► ►►►

$p_n(x)$	$w(x)$	$F(x)$	${\kappa }_n$
…
$C_n^{(\lambda )}\left(x\right)$	${(1-x^2)}^{\lambda -\frac{1}{2}}$	$1-x^2$	${\displaystyle \frac{{(-2)}^n{\left(\lambda +\frac{1}{2}\right)}_{n}n!}{{\left(2\lambda \right)}_n}}$
…

►

… ►See (Erdélyi et al., 1953b, §10.9(37)) for a related formula for ultraspherical polynomials. … ►

18.5.9 $$C_n^{(\lambda )}\left(x\right)=\frac{{\left(2\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,n+2\lambda }{\lambda +\frac{1}{2}};\frac{1-x}{2}),$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.46

Referenced by:

§18.5(iii), §18.6(i)

Permalink:

http://dlmf.nist.gov/18.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►

18.5.11 $$C_n^{(\lambda )}\left(\cos \theta \right)=\sum _{\mathrm{\ell }=0}^n\frac{{\left(\lambda \right)}_{\mathrm{\ell }}{\left(\lambda \right)}_{n-\mathrm{\ell }}}{\mathrm{\ell }!(n-\mathrm{\ell })!}\cos \left((n-2\mathrm{\ell })\theta \right)={\mathrm{e}}^{\mathrm{i}n\theta }\frac{{\left(\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,\lambda }{1-\lambda -n};{\mathrm{e}}^{-2\mathrm{i}\theta }).$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

22.3.12

Referenced by:

§18.18(iv), §18.5(iii)

Permalink:

http://dlmf.nist.gov/18.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

… ►Similarly in the cases of the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$ and the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ we assume that $\lambda >-\frac{1}{2},\lambda \ne 0$, and $\alpha >-1$, unless stated otherwise. …

13: 1.10 Functions of a Complex Variable

… ►Ultraspherical polynomials have generating function ►

1.10.28 $$F(x,\lambda ;z)={\left(1-2xz+z^2\right)}^{-\lambda }=\sum _{n=0}^{\mathrm{\infty }}{C}_n^{(\lambda )}\left(x\right)z^n,$$ .

ⓘ

Symbols:

$C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $z$: variable, $n$: nonnegative integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.10.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(xi), §1.10(xi), §1.10 and Ch.1

… ►

1.10.29 $$\sum _{n=0}^{\mathrm{\infty }}\frac{d}{dx}{C}_n^{(\lambda )}\left(x\right)z^n=2\lambda z{\left(1-2xz+z^2\right)}^{-\lambda -1}=\sum _{n=0}^{\mathrm{\infty }}2\lambda C_n^{(\lambda +1)}\left(x\right)z^{n+1},$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $z$: variable and $n$: nonnegative integer

Referenced by:

Erratum (V1.2.0) Section 1.10

Permalink:

http://dlmf.nist.gov/1.10.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §1.10(xi), §1.10(xi), §1.10 and Ch.1

►and hence $\frac{d}{dx}{C}_n^{(\lambda )}\left(x\right)=2\lambda C_{n-1}^{(\lambda +1)}\left(x\right)$, that is (18.9.19). The recurrence relation for $C_n^{(\lambda )}\left(x\right)$ in §18.9(i) follows from $(1-2xz+z^2)\frac{\partial }{\partial z}F(x,\lambda ;z)=2\lambda (x-z)F(x,\lambda ;z)$, and the contour integral representation for $C_n^{(\lambda )}\left(x\right)$ in §18.10(iii) is just (1.10.27).

14: 10.23 Sums

… ►

10.23.8 $$\frac{{𝒞}_{\nu }\left(w\right)}{w^{\nu }}=2^{\nu }\mathrm{\Gamma }\left(\nu \right)\sum _{k=0}^{\mathrm{\infty }}(\nu +k)\frac{{𝒞}_{\nu +k}\left(u\right)}{u^{\nu }}\frac{J_{\nu +k}\left(v\right)}{v^{\nu }}{C}_k^{(\nu )}\left(\cos \alpha \right),$$ $\nu \ne 0,-1,\mathrm{\dots }$, ,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\cos z$: cosine function, ${𝒞}_{\nu }\left(z\right)$: cylinder function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $k$: nonnegative integer and $\nu$: complex parameter

A&S Ref:

9.1.80

Referenced by:

§10.23(ii), §10.23(ii), §10.23(ii), §10.44(ii), §10.60(i), §10.60(ii)

Permalink:

http://dlmf.nist.gov/10.23.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

►where $C_k^{(\nu )}\left(\cos \alpha \right)$ is Gegenbauer’s polynomial (§18.3). … ►

10.23.9 $${\mathrm{e}}^{\mathrm{i}v\cos \alpha }=\frac{\mathrm{\Gamma }\left(\nu \right)}{{(\frac{1}{2}v)}^{\nu }}\sum _{k=0}^{\mathrm{\infty }}(\nu +k){\mathrm{i}}^k{J}_{\nu +k}\left(v\right)C_k^{(\nu )}\left(\cos \alpha \right),$$ $\nu \ne 0,-1,\mathrm{\dots }$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $k$: nonnegative integer and $\nu$: complex parameter

A&S Ref:

9.1.81

Referenced by:

§10.23(ii)

Permalink:

http://dlmf.nist.gov/10.23.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.23(ii), §10.23(ii), §10.23 and Ch.10

…

15: 18.11 Relations to Other Functions

… ►

Ultraspherical

►

18.11.1 $${𝖯}_n^m\left(x\right)={\left(\frac{1}{2}\right)}_m{(-2)}^m{(1-x^2)}^{\frac{1}{2}m}{C}_{n-m}^{(m+\frac{1}{2})}\left(x\right)={\left(n+1\right)}_m{(-2)}^{-m}{(1-x^2)}^{\frac{1}{2}m}{P}_{n-m}^{(m,m)}\left(x\right),$$ $0\le m\le n$.

ⓘ

Symbols:

: Ferrers function of the first kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§14.3(iv), §18.11(i)

Permalink:

http://dlmf.nist.gov/18.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.11(i), §18.11(i), §18.11 and Ch.18

…

16: 15.9 Relations to Other Functions

… ►

Gegenbauer (or Ultraspherical)

►

15.9.2 $$C_n^{(\lambda )}\left(x\right)=\frac{{\left(2\lambda \right)}_n}{n!}F(\genfrac{}{}{0pt}{}{-n,n+2\lambda }{\lambda +\frac{1}{2}};\frac{1-x}{2}).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

►

15.9.3 $$C_n^{(\lambda )}\left(x\right)={(2x)}^n\frac{{\left(\lambda \right)}_n}{n!}F(\genfrac{}{}{0pt}{}{-\frac{1}{2}n,\frac{1}{2}(1-n)}{1-\lambda -n};\frac{1}{x^2}).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $x$: real variable and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

►

15.9.4 $$C_n^{(\lambda )}\left(\cos \theta \right)={\mathrm{e}}^{n\mathrm{i}\theta }\frac{{\left(\lambda \right)}_n}{n!}F(\genfrac{}{}{0pt}{}{-n,\lambda }{1-\lambda -n};{\mathrm{e}}^{-2\mathrm{i}\theta }).$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial and $n$: integer

Permalink:

http://dlmf.nist.gov/15.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(i), §15.9(i), §15.9 and Ch.15

… ►This is a generalization of Gegenbauer (or ultraspherical) polynomials (§18.3). …

17: 18.28 Askey–Wilson Class

… ►

§18.28(v) Continuous $q$-Ultraspherical Polynomials

►

18.28.13 $$C_n(\cos \theta ;\beta |q)=\sum _{\mathrm{\ell }=0}^n\frac{{(\beta ;q)}_{\mathrm{\ell }}{(\beta ;q)}_{n-\mathrm{\ell }}}{{(q;q)}_{\mathrm{\ell }}{(q;q)}_{n-\mathrm{\ell }}}{\mathrm{e}}^{\mathrm{i}(n-2\mathrm{\ell })\theta }=\frac{{(\beta ;q)}_n}{{(q;q)}_n}{\mathrm{e}}^{\mathrm{i}n\theta }{}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{-n},\beta }{{\beta }^{-1}{q}^{1-n}};q,{\beta }^{-1}q{\mathrm{e}}^{-2\mathrm{i}\theta }).$$

ⓘ

Defines:

$C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial

Symbols:

$\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: real variable, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.28.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(v), §18.28 and Ch.18

… ►

18.28.25 $$P_n^{(\lambda -\frac{1}{2},\lambda -\frac{1}{2})}(x|q)=\frac{q^{\frac{1}{2}n\lambda }{(q^{\lambda +1/2};q)}_n}{{(q^{2\lambda };q)}_n}{C}_n(x;q^{\lambda }|q).$$

ⓘ

Symbols:

$(a,b)$: open interval, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper and Rahman (2004, (7.5.33))(proved)

Permalink:

http://dlmf.nist.gov/18.28.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(ix), §18.28 and Ch.18

… ►

18.28.31 $$\underset{q\to 1}{lim}{C}_n(x;q^{\lambda }|q)=C_n^{(\lambda )}\left(x\right).$$

ⓘ

Symbols:

$C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.10.35))

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

… ►

18.28.32 $$\underset{\beta \to 0}{lim}{C}_n(x;\beta |q)=\frac{H_n\left(x|q\right)}{{(q;q)}_n}.$$

ⓘ

Symbols:

$H_n\left(x|q\right)$: continuous $q$-Hermite polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $C_n(x;\beta |q)$: continuous $q$-ultraspherical polynomial, $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.10.34))

Referenced by:

§18.28(x)

Permalink:

http://dlmf.nist.gov/18.28.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

…

18: 18.35 Pollaczek Polynomials

… ►The type 2 polynomials reduce for $a=b=0$ to ultraspherical polynomials, see (18.35.8). … ►

18.35.8 $$P_n^{(\lambda )}(x;0,0)=C_n^{(\lambda )}\left(x\right),$$

ⓘ

Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable

Proved:

Ismail (2009, (5.4.3))(proved)

Referenced by:

§18.30(viii), §18.35(i)

Permalink:

http://dlmf.nist.gov/18.35.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.35(iii), §18.35 and Ch.18

… ►For the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$, the Meixner–Pollaczek polynomials $P_n^{(\lambda )}(x;\varphi )$ and the associated Meixner–Pollaczek polynomials ${𝒫}_n^{\lambda }(x;\varphi ,c)$ see §§18.3, 18.19 and 18.30(v), respectively. …

19: 18.15 Asymptotic Approximations

… ►

§18.15(ii) Ultraspherical

… ►

18.15.10 $$C_n^{(\lambda )}\left(\cos \theta \right)=\frac{2^{2\lambda }\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\lambda +1\right)}\frac{{\left(2\lambda \right)}_n}{{\left(\lambda +1\right)}_n}\left(\sum _{m=0}^{M-1}\frac{{\left(\lambda \right)}_m{\left(1-\lambda \right)}_m}{m!{\left(n+\lambda +1\right)}_m}\frac{\cos {\theta }_{n,m}}{{(2\sin \theta )}^{m+\lambda }}+O\left(\frac{1}{n^M}\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $m$: nonnegative integer, $n$: nonnegative integer and ${\theta }_{n,m}$

Referenced by:

§18.15(ii), §18.15(iii), §18.15(iii)

Permalink:

http://dlmf.nist.gov/18.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(ii), §18.15 and Ch.18

… ►Asymptotic expansions for $C_n^{(\lambda )}\left(\cos \theta \right)$ can be obtained from the results given in §18.15(i) by setting $\alpha =\beta =\lambda -\frac{1}{2}$ and referring to (18.7.1). … ►For asymptotic approximations of Jacobi, ultraspherical, and Laguerre polynomials in terms of Hermite polynomials, see López and Temme (1999a). These approximations apply when the parameters are large, namely $\alpha$ and $\beta$ (subject to restrictions) in the case of Jacobi polynomials, $\lambda$ in the case of ultraspherical polynomials, and $|\alpha |+|x|$ in the case of Laguerre polynomials. …

20: 10.60 Sums

… ►Then with $P_n$ again denoting the Legendre polynomial of degree $n$, ►

10.60.1 $$\frac{\cos w}{w}=-\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗃}_n\left(v\right){𝗒}_n\left(u\right)P_n\left(\cos \alpha \right),$$ .

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, ${𝗒}_n\left(z\right)$: spherical Bessel function of the second kind and $n$: integer

A&S Ref:

10.1.46 (corrected)

Referenced by:

§10.60(i)

Permalink:

http://dlmf.nist.gov/10.60.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(i), §10.60 and Ch.10

►

10.60.2 $$\frac{\sin w}{w}=\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗃}_n\left(v\right){𝗃}_n\left(u\right)P_n\left(\cos \alpha \right).$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\sin z$: sine function, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer

A&S Ref:

10.1.45

Referenced by:

§10.60(iii)

Permalink:

http://dlmf.nist.gov/10.60.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(i), §10.60 and Ch.10

… ►

10.60.7 $${\mathrm{e}}^{\mathrm{i}z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}(2n+1){\mathrm{i}}^n{𝗃}_n\left(z\right)P_n\left(\cos \alpha \right),$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

A&S Ref:

10.1.47

Referenced by:

§10.60(iii), Ch.10

Permalink:

http://dlmf.nist.gov/10.60.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(iii), §10.60 and Ch.10

►

10.60.8 $${\mathrm{e}}^{z\cos \alpha }=\sum _{n=0}^{\mathrm{\infty }}(2n+1){𝗂}_n^{(1)}\left(z\right)P_n\left(\cos \alpha \right),$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

A&S Ref:

10.2.36

Permalink:

http://dlmf.nist.gov/10.60.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.60(iii), §10.60 and Ch.10

…