Gegenbauer%20addition%20theorem

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11: 18.17 Integrals

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18.17.5 $$\frac{C_n^{(\lambda )}\left(\cos {\theta }_1\right)}{C_n^{(\lambda )}\left(1\right)}\frac{C_n^{(\lambda )}\left(\cos {\theta }_2\right)}{C_n^{(\lambda )}\left(1\right)}=\frac{\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\lambda \right)}{\int }_0^{\pi }\frac{C_n^{(\lambda )}\left(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \varphi \right)}{C_n^{(\lambda )}\left(1\right)}{(\sin \varphi )}^{2\lambda -1}d\varphi ,$$ $\lambda >0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial and $n$: nonnegative integer

Referenced by:

§18.17(ii), §18.17(ii), §18.18(ii)

Permalink:

http://dlmf.nist.gov/18.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(ii), §18.17(ii), §18.17 and Ch.18

… ►For addition formulas corresponding to (18.17.5) and (18.17.6) see (18.18.8) and (18.18.9), respectively. … ►

18.17.12 $$\frac{\mathrm{\Gamma }\left(\lambda -\mu \right)C_n^{(\lambda -\mu )}\left(x^{-\frac{1}{2}}\right)}{x^{\lambda -\mu +\frac{1}{2}n}}={\int }_x^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\lambda \right)C_n^{(\lambda )}\left(y^{-\frac{1}{2}}\right)}{y^{\lambda +\frac{1}{2}n}}\frac{{(y-x)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\lambda >\mu >0$, $x>0$,

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Symbols:

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

Referenced by:

§18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

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18.17.13 $$\frac{x^{\frac{1}{2}n}{(x-1)}^{\lambda +\mu -\frac{1}{2}}}{\mathrm{\Gamma }\left(\lambda +\mu +\frac{1}{2}\right)}\frac{C_n^{(\lambda +\mu )}\left(x^{-\frac{1}{2}}\right)}{C_n^{(\lambda +\mu )}\left(1\right)}={\int }_1^x\frac{y^{\frac{1}{2}n}{(y-1)}^{\lambda -\frac{1}{2}}}{\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}\frac{C_n^{(\lambda )}\left(y^{-\frac{1}{2}}\right)}{C_n^{(\lambda )}\left(1\right)}\frac{{(x-y)}^{\mu -1}}{\mathrm{\Gamma }\left(\mu \right)}dy,$$ $\mu >0$, $x>1$.

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Symbols:

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

Referenced by:

§1.15(vi), §18.17(iv)

Permalink:

http://dlmf.nist.gov/18.17.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(iv), §18.17(iv), §18.17 and Ch.18

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18.17.16_5 $${\int }_{-1}^1{(1-x^2)}^{\lambda -\frac{1}{2}}{C}_n^{(\lambda )}\left(x\right){\mathrm{e}}^{\mathrm{i}xy}dx=\frac{2\pi {\mathrm{i}}^n\mathrm{\Gamma }\left(n+2\lambda \right)J_{n+\lambda }\left(y\right)}{n!\mathrm{\Gamma }\left(\lambda \right){(2y)}^{\lambda }},$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\int$: integral, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $y$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.17(v), Erratum (V1.2.0) Section 18.17

Permalink:

http://dlmf.nist.gov/18.17.E16_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.17(v), §18.17(v), §18.17 and Ch.18

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12: 1.10 Functions of a Complex Variable

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Picard’s Theorem

… ►In addition, … ►Also, if in addition $g(z)$ is analytic at $z_0$, then … ►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).

13: 18.5 Explicit Representations

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

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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

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18.5.10 $$C_n^{(\lambda )}\left(x\right)=\sum _{\mathrm{\ell }=0}^{\lfloor n/2\rfloor }\frac{{(-1)}^{\mathrm{\ell }}{\left(\lambda \right)}_{n-\mathrm{\ell }}}{\mathrm{\ell }!(n-2\mathrm{\ell })!}{(2x)}^{n-2\mathrm{\ell }}={(2x)}^n\frac{{\left(\lambda \right)}_n}{n!}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}n,-\frac{1}{2}n+\frac{1}{2}}{1-\lambda -n};\frac{1}{x^2}),$$

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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 $n!$), $\lfloor x\rfloor$: floor of $x$ , $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.3.4

Referenced by:

§18.17(vii), §18.5(iii), §18.6(ii)

Permalink:

http://dlmf.nist.gov/18.5.E10

Encodings:

pMML, png, TeX

See also:

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

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

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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. … ►

$T_5\left(x\right)=16x^5-20x^3+5x,$

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14: Tom M. Apostol

… ►Apostol was born on August 20, 1923. … ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). …In addition, he was the co-author of New Horizons in Geometry, published by the MAA, which received the CHOICE “Outstanding Academic Title” award in 2013. …He additionally served as a visiting lecturer for the MAA, and as a member of the MAA Board of Governors. …

15: 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

… ►

§15.9(iii) Gegenbauer Function

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

16: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product $m$ has 20 digits, twice the number of digits in the data. By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

17: 14.28 Sums

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§14.28(i) Addition Theorem

… ►For generalizations in terms of Gegenbauer and Jacobi polynomials, see Theorem 2. 1 in Cohl (2013b) and Theorem 1 in Cohl (2013a) respectively. …

18: 18.14 Inequalities

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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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18.14.26 $$\sum _{m=0}^n\frac{P_m^{(\alpha ,\beta )}\left(x\right)}{P_m^{(\beta ,\alpha )}\left(1\right)}\ge 0,$$ $x\ge -1$, $n=0,1,\mathrm{\dots }$,

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Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Gasper (1977, Theorem 4); also proved there for $\alpha +\beta \ge -2$, $\beta \ge 0$(proved)

Source:

Askey and Gasper (1976, (1.7), (1.12)); conjectured

Referenced by:

§18.14(iv), §18.38(ii)

Permalink:

http://dlmf.nist.gov/18.14.E26

Encodings:

pMML, png, TeX

See also:

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

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19: 18.3 Definitions

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As eigenfunctions of second order differential operators (Bochner’s theorem, Bochner (1929)). See the differential equations $A(x)p_n^{\prime \prime }(x)+B(x)p_n^{\prime }(x)+{\lambda }_n{p}_n(x)=0$, in Table 18.8.1.

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Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
…
Ultraspherical (Gegenbauer)	$C_n^{(\lambda )}\left(x\right)$	$(-1,1)$	${(1-x^2)}^{\lambda -\frac{1}{2}}$	${\displaystyle \frac{2^{1-2\lambda }\pi \mathrm{\Gamma }\left(n+2\lambda \right)}{(n+\lambda ){\left(\mathrm{\Gamma }\left(\lambda \right)\right)}^{2}n!}}$	${\displaystyle \frac{2^n{\left(\lambda \right)}_n}{n!}}$	$0$	$\lambda >-\frac{1}{2},\lambda \ne 0$
…

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… ►In addition to the orthogonal property given by Table 18.3.1, the Chebyshev polynomials $T_n\left(x\right)$, $n=0,1,\mathrm{\dots },N$, are orthogonal on the discrete point set comprising the zeros $x_{N+1,n},n=1,2,\mathrm{\dots },N+1$, of $T_{N+1}\left(x\right)$: …

20: Errata

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18.7.25

We included the case $n=0$.

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The Gegenbauer function $C_{\alpha }^{(\lambda )}\left(z\right)$, was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial $C_n^{(\lambda )}\left(z\right)$. In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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The coefficient $A_n$ for $C_n^{(\lambda )}\left(x\right)$ in the first row of this table originally omitted the parentheses and was given as $\frac{2n+\lambda }{n+1}$, instead of $\frac{2(n+\lambda )}{n+1}$.

$p_n(x)$	$A_n$	$B_n$	$C_n$
$C_n^{(\lambda )}\left(x\right)$	$\frac{2(n+\lambda )}{n+1}$	$0$	$\frac{n+2\lambda -1}{n+1}$
⋮

Reported 2010-09-16 by Kendall Atkinson.

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Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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