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11: 18.14 Inequalities

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Ultraspherical

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

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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

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

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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

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

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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

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

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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

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12: 18.5 Explicit Representations

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

§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

… ►

Ultraspherical

… ►For corresponding formulas for Chebyshev, Legendre, and the Hermite ${\mathrm{𝐻𝑒}}_n$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ►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: 18.11 Relations to Other Functions

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Ultraspherical

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

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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

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14: 15.9 Relations to Other Functions

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Gegenbauer (or Ultraspherical)

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

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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

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

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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

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

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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). …

15: 1.10 Functions of a Complex Variable

… ►Ultraspherical polynomials have generating function ►

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

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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

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

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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).

16: 10.23 Sums

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

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

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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

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17: 18.28 Askey–Wilson Class

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

… ►Specialization to continuous $q$-ultraspherical: … ►

From Continuous $q$-Ultraspherical to Ultraspherical

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

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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

►

From Continuous $q$-Ultraspherical to Continuous $q$-Hermite

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18: 18.15 Asymptotic Approximations

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§18.15(ii) Ultraspherical

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

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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. …

19: 14.3 Definitions and Hypergeometric Representations

… ►In terms of the Gegenbauer function $C_{\alpha }^{(\beta )}\left(x\right)$ and the Jacobi function ${\varphi }_{\lambda }^{(\alpha ,\beta )}\left(t\right)$ (§§15.9(iii), 15.9(ii)): ►

14.3.21

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $C_{\alpha }^{(\lambda )}\left(z\right)$: Gegenbauer function, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.3(iv), §14.3(iv), §18.10(i), §18.11(i)

Permalink:

http://dlmf.nist.gov/14.3.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iv), §14.3 and Ch.14

►

14.3.22

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $C_{\alpha }^{(\lambda )}\left(z\right)$: Gegenbauer function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\mu$: general order, $\nu$: general degree and : real variable

Referenced by:

§14.3(iv), §14.3(iv), §18.11(i)

Permalink:

http://dlmf.nist.gov/14.3.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §14.3(iv), §14.3 and Ch.14

…

20: 18.35 Pollaczek Polynomials

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

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

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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. …