Jacobi function

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31: 20.10 Integrals

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20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_2\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_4\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.2 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}({\theta }_3\left(0\right|\mathrm{i}{x}^2)-1)dx={\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E2

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_3\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_3\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.32) and (20.4.4).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.3 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}(1-{\theta }_4\left(0\right|\mathrm{i}{x}^2))dx=(1-2^{1-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>0$ because $1-{\theta }_4\left(0\right|\mathrm{i}{x}^2)=1-x^{-1}{\theta }_2\left(0\right|\mathrm{i}{x}^{-2})\sim 1$ as $x\to 0+$, which follows from (20.7.33) and (20.4.3).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

…

32: 29.15 Fourier Series and Chebyshev Series

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29.15.45 $${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)=\mathrm{cn}(z,k)\sum _{p=0}^n{B}_{2p+1}{U}_{2p}\left(\mathrm{sn}(z,k)\right),$$

ⓘ

Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑐𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $B_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(ii), §29.15 and Ch.29

►

29.15.46 $${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)=\mathrm{dn}(z,k)\left(\frac{1}{2}{C}_0+\sum _{p=1}^n{C}_{2p}{T}_{2p}\left(\mathrm{sn}(z,k)\right)\right),$$

ⓘ

Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑑𝐸}}_{2n+1}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $C_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E46

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(ii), §29.15 and Ch.29

►

29.15.47 $${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)=\mathrm{cn}(z,k)\sum _{p=0}^n{B}_{2p+2}{U}_{2p+1}\left(\mathrm{sn}(z,k)\right),$$

ⓘ

Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑠𝑐𝐸}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $B_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E47

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(ii), §29.15 and Ch.29

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29.15.49 $${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)=\mathrm{cn}(z,k)\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+1}{U}_{2p}\left(\mathrm{sn}(z,k)\right),$$

ⓘ

Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑐𝑑𝐸}}_{2n+2}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $D_{2p}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(ii), §29.15 and Ch.29

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29.15.50 $${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)=\mathrm{cn}(z,k)\mathrm{dn}(z,k)\sum _{p=0}^n{D}_{2p+2}{U}_{2p+1}\left(\mathrm{sn}(z,k)\right).$$

ⓘ

Symbols:

$U_n\left(x\right)$: Chebyshev polynomial of the second kind, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝑠𝑐𝑑𝐸}}_{2n+3}^m(z,k^2)$: Lamé polynomial, $m$: nonnegative integer, $n$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter and $D_{2p+1}$: coefficents

Permalink:

http://dlmf.nist.gov/29.15.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §29.15(ii), §29.15 and Ch.29

…

33: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation … ►

20.13.4 $$\sqrt{\frac{\pi }{4\alpha t}}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-{(n\pi +z)}^2/(4\alpha t)}={\theta }_3\left(z\right|\mathrm{i}4\alpha t/\pi ),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: integer, $t$: time, $\alpha$: real parameter and $z$: real variable

Permalink:

http://dlmf.nist.gov/20.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

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20.13.5 $$\sqrt{\frac{\pi }{4\alpha t}}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{\mathrm{e}}^{-{(n\pi +z)}^2/(4\alpha t)}={\theta }_4\left(z\right|\mathrm{i}4\alpha t/\pi ).$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: integer, $t$: time, $\alpha$: real parameter and $z$: real variable

Permalink:

http://dlmf.nist.gov/20.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

… ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …

34: 20.15 Tables

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20.15.1 $$\sin \alpha ={\theta }_2^2(0,q)/{\theta }_3^2(0,q)=k.$$

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Defines:

$\alpha$: modular angle (locally)

Symbols:

${\theta }_j(z,q)$: theta function, $\sin z$: sine function and $q$: nome

Referenced by:

§20.15, §20.15

Permalink:

http://dlmf.nist.gov/20.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.15 and Ch.20

… ►Tables of Neville’s theta functions ${\theta }_s(x,q)$, ${\theta }_c(x,q)$, ${\theta }_d(x,q)$, ${\theta }_n(x,q)$ (see §20.1) and their logarithmic $x$-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for $\epsilon ,\alpha =0(5^{\circ }){90}^{\circ }$, where (in radian measure) $\epsilon =x/{\theta }_3^2(0,q)=\pi x/(2K\left(k\right))$, and $\alpha$ is defined by (20.15.1). …

35: 20.9 Relations to Other Functions

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20.9.1 $$k={\theta }_2^2\left(0\right|\tau )/{\theta }_3^2\left(0\right|\tau )$$

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Defines:

$k$: modulus (locally)

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function and $\tau$: lattice parameter

Referenced by:

§20.11(iii), §20.9(ii), §23.6(ii)

Permalink:

http://dlmf.nist.gov/20.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.9(i), §20.9 and Ch.20

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20.9.3 $$R_F(\frac{{\theta }_2^2(z,q)}{{\theta }_2^2(0,q)},\frac{{\theta }_3^2(z,q)}{{\theta }_3^2(0,q)},\frac{{\theta }_4^2(z,q)}{{\theta }_4^2(0,q)})=\frac{{\theta }_1^{\prime }(0,q)}{{\theta }_1(z,q)}z,$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, ${\theta }_j(z,q)$: theta function, $z$: complex and $q$: nome

Referenced by:

§19.25(v), §20.9(i)

Permalink:

http://dlmf.nist.gov/20.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.9(i), §20.9 and Ch.20

►

20.9.4 $$R_F(0,{\theta }_3^4(0,q),{\theta }_4^4(0,q))=\frac{1}{2}\pi ,$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, ${\theta }_j(z,q)$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter and $q$: nome

Referenced by:

§20.9(i)

Permalink:

http://dlmf.nist.gov/20.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.9(i), §20.9 and Ch.20

… ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …

36: 29.8 Integral Equations

… ►

29.8.1 $$x=k^2\mathrm{sn}(z,k)\mathrm{sn}(z_1,k)\mathrm{sn}(z_2,k)\mathrm{sn}(z_3,k)-\frac{k^2}{k_{}^{\prime }{}_{}{}^2}\mathrm{cn}(z,k)\mathrm{cn}(z_1,k)\mathrm{cn}(z_2,k)\mathrm{cn}(z_3,k)+\frac{1}{k_{}^{\prime }{}_{}{}^2}\mathrm{dn}(z,k)\mathrm{dn}(z_1,k)\mathrm{dn}(z_2,k)\mathrm{dn}(z_3,k),$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $x$: real variable, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

►where $z,z_1,z_2,z_3$ are real, and $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ are the Jacobian elliptic functions (§22.2). … ►

29.8.6 $$y=\frac{1}{k^{\prime }}\mathrm{dn}(z,k)\mathrm{dn}(z_1,k).$$

ⓘ

Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $y$: real variable, $z$: complex variable and $k$: real parameter

Permalink:

http://dlmf.nist.gov/29.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

… ►

29.8.7 $${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z_1,k^2)\frac{w_2(K)+w_2(-K)}{w_2(0)}=-k^2\mathrm{sn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\frac{d{𝖯}_{\nu }\left(y\right)}{dy}{\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)dz,$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: 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.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

… ►

29.8.9 $${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z_1,k^2)\frac{{d{w}_2(z)/dz|}_{z=K}-{d{w}_2(z)/dz|}_{z=-K}}{w_2(0)}=-\frac{k^4}{k^{\prime }}\mathrm{sn}(z_1,k)\mathrm{cn}(z_1,k){\int }_{-K}^K\mathrm{sn}(z,k)\mathrm{cn}(z,k)\frac{d^2{𝖯}_{\nu }\left(y\right)}{{dy}^2}{\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)dz.$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: 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.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §29.8 and Ch.29

…

37: 18.12 Generating Functions

… ►

Jacobi

… ►

18.12.2 $${}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\alpha +1};\frac{(x-1)z}{2}){}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\beta +1};\frac{(x+1)z}{2})={\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n,$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (9.8.12))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) for Equation (18.12.2)

Permalink:

http://dlmf.nist.gov/18.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

►

18.12.2_5 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\alpha +1};\frac{1-R-z}{2}){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\beta +1};\frac{1-R+z}{2})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\gamma \right)}_n{\left(\alpha +\beta +1-\gamma \right)}_n}{{\left(\alpha +1\right)}_n{\left(\beta +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(x\right)z^n,$$ $R=\sqrt{1-2xz+z^2}$, ,

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proved:

Rainville (1960, §141, (2)); Brafman’s generating function(proved)

Source:

Koekoek et al. (2010, (9.8.15))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) Section 18.12

Permalink:

http://dlmf.nist.gov/18.12.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

… ►

18.12.3 $${(1+z)}^{-\alpha -\beta -1}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}(\alpha +\beta +1),\frac{1}{2}(\alpha +\beta +2)}{\beta +1};\frac{2(x+1)z}{{(1+z)}^2})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\alpha +\beta +1\right)}_n}{{\left(\beta +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(x\right)z^n,$$ ,

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.12.3_5), §18.12, §18.12

Permalink:

http://dlmf.nist.gov/18.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

…

38: 18.38 Mathematical Applications

… ►The Askey–Gasper inequality ►

18.38.3 $$\sum _{m=0}^n{P}_m^{(\alpha ,0)}\left(x\right)=\frac{{\left(\alpha +2\right)}_n}{n!}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-n,n+\alpha +2,\frac{1}{2}(\alpha +1)}{\alpha +1,\frac{1}{2}(\alpha +3)};\frac{1}{2}(1-x))\ge 0,$$ $x\ge -1$, $\alpha \ge -2$, $n=0,1,\mathrm{\dots }$,

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Askey and Gasper (1976, Theorem 3, p.717)(proved)

Referenced by:

§18.14(iv), §18.18(viii), Erratum (V1.2.0) for Equation (18.38.3)

Permalink:

http://dlmf.nist.gov/18.38.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation was updated to include the value of the sum in terms of the ${}_3{}^{}F_2^{}$ function. Also the constraint was previously $\alpha >-1$.

See also:

Annotations for §18.38(ii), §18.38(ii), §18.38 and Ch.18

►also the case $\beta =0$ of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. …

39: 18.10 Integral Representations

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18.10.1 $$\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{2^{\alpha +\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha +\frac{1}{2}\right)}{(\sin \theta )}^{-2\alpha }{\int }_0^{\theta }\frac{\cos \left((n+\alpha +\frac{1}{2})\varphi \right)}{{(\cos \varphi -\cos \theta )}^{-\alpha +\frac{1}{2}}}d\varphi ,$$ , $\alpha >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: 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

A&S Ref:

22.10.11

Referenced by:

§18.10(i), §18.10(i)

Permalink:

http://dlmf.nist.gov/18.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(i), §18.10(i), §18.10 and Ch.18

… ►

18.10.3 $$\frac{P_n^{(\alpha ,\beta )}\left(\cos \theta \right)}{P_n^{(\alpha ,\beta )}\left(1\right)}=\frac{2\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\alpha -\beta \right)\mathrm{\Gamma }\left(\beta +\frac{1}{2}\right)}{\int }_0^1{\int }_0^{\pi }{\left({(\cos \frac{1}{2}\theta )}^2-r^2{(\sin \frac{1}{2}\theta )}^2+ir\sin \theta \cos \varphi \right)}^n{(1-r^2)}^{\alpha -\beta -1}{r}^{2\beta +1}{(\sin \varphi )}^{2\beta }d\varphi dr,$$ $\alpha >\beta >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: 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 and $n$: nonnegative integer

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

… ►

18.10.4 $$\frac{P_n^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}\left(\cos \theta \right)}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{\mathrm{\Gamma }\left(\alpha +1\right)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }(\alpha +\frac{1}{2})}{\int }_0^{\pi }{(\cos \theta +i\sin \theta \cos \varphi )}^n{(\sin \varphi )}^{2\alpha }d\varphi ,$$ $\alpha >-\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $\pi$: 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

A&S Ref:

22.10.10

Referenced by:

§18.10(ii)

Permalink:

http://dlmf.nist.gov/18.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(ii), §18.10(ii), §18.10 and Ch.18

…

40: 20.5 Infinite Products and Related Results

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20.5.1 $${\theta }_1(z,q)=2q^{1/4}\sin z\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1-2q^{2n}\cos \left(2z\right)+q^{4n}\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $\sin z$: sine function, $n$: integer, $z$: complex and $q$: nome

Referenced by:

§20.4(i), §20.5(i), §23.17(iii)

Permalink:

http://dlmf.nist.gov/20.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

►

20.5.2 $${\theta }_2(z,q)=2q^{1/4}\cos z\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+2q^{2n}\cos \left(2z\right)+q^{4n}\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

►

20.5.3 $${\theta }_3(z,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1+2q^{2n-1}\cos \left(2z\right)+q^{4n-2}\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex and $q$: nome

Referenced by:

§23.15(ii), §23.17(iii), §23.19

Permalink:

http://dlmf.nist.gov/20.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

►

20.5.4 $${\theta }_4(z,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right)\left(1-2q^{2n-1}\cos \left(2z\right)+q^{4n-2}\right).$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex and $q$: nome

Referenced by:

§20.4(i), §20.5(i)

Permalink:

http://dlmf.nist.gov/20.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.5(i), §20.5 and Ch.20

… ►

Jacobi’s Triple Product

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