Watson 3F2 sum

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11: 20.4 Values at $z$ = 0

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20.4.2 $${\theta }_1^{\prime }(0,q)=2q^{1/4}\prod _{n=1}^{\mathrm{\infty }}{\left(1-q^{2n}\right)}^3=2q^{1/4}{(q^2;q^2)}_{\mathrm{\infty }}^3,$$

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Symbols:

${\theta }_j(z,q)$: theta function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $n$: integer and $q$: nome

Referenced by:

Erratum (V1.0.28) for Equation (20.4.2)

Permalink:

http://dlmf.nist.gov/20.4.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.28):

The representation in terms of ${(q^2;q^2)}_{\mathrm{\infty }}^3$ was added to this equation.

See also:

Annotations for §20.4(i), §20.4 and Ch.20

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20.4.4 $${\theta }_3(0,q)=\prod _{n=1}^{\mathrm{\infty }}\left(1-q^{2n}\right){\left(1+q^{2n-1}\right)}^2,$$

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Symbols:

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

Referenced by:

(20.10.2)

Permalink:

http://dlmf.nist.gov/20.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4 and Ch.20

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20.4.6 $${\theta }_1^{\prime }(0,q)={\theta }_2(0,q){\theta }_3(0,q){\theta }_4(0,q).$$

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Symbols:

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

A&S Ref:

16.28.6

Referenced by:

§20.9(i)

Permalink:

http://dlmf.nist.gov/20.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(i), §20.4(i), §20.4 and Ch.20

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20.4.10 $$\frac{{\theta }_3^{\prime \prime }(0,q)}{{\theta }_3(0,q)}=-8\sum _{n=1}^{\mathrm{\infty }}\frac{q^{2n-1}}{{(1+q^{2n-1})}^2},$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/20.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(ii), §20.4 and Ch.20

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20.4.12 $$\frac{{\theta }_1^{\prime \prime \prime }(0,q)}{{\theta }_1^{\prime }(0,q)}=\frac{{\theta }_2^{\prime \prime }(0,q)}{{\theta }_2(0,q)}+\frac{{\theta }_3^{\prime \prime }(0,q)}{{\theta }_3(0,q)}+\frac{{\theta }_4^{\prime \prime }(0,q)}{{\theta }_4(0,q)}.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/20.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §20.4(ii), §20.4 and Ch.20

12: 23.3 Differential Equations

… ►and are denoted by $e_1,e_2,e_3$. … ►Let $g_2^3\ne 27g_3^2$, or equivalently $\mathrm{\Delta }$ be nonzero, or $e_1,e_2,e_3$ be distinct. …Similarly for $\zeta (z;g_2,g_3)$ and $\sigma (z;g_2,g_3)$. As functions of $g_2$ and $g_3$, $\mathrm{\wp }(z;g_2,g_3)$ and $\zeta (z;g_2,g_3)$ are meromorphic and $\sigma (z;g_2,g_3)$ is entire. ►Conversely, $g_2$, $g_3$, and the set $\{e_1,e_2,e_3\}$ are determined uniquely by the lattice $𝕃$ independently of the choice of generators. …

13: 10.31 Power Series

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10.31.1 $$K_n\left(z\right)=\frac{1}{2}{(\frac{1}{2}z)}^{-n}\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}{(-\frac{1}{4}{z}^2)}^k+{(-1)}^{n+1}\ln \left(\frac{1}{2}z\right)I_n\left(z\right)+{(-1)}^n\frac{1}{2}{(\frac{1}{2}z)}^n\sum _{k=0}^{\mathrm{\infty }}\left(\psi \left(k+1\right)+\psi \left(n+k+1\right)\right)\frac{{(\frac{1}{4}{z}^2)}^k}{k!(n+k)!},$$

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Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.6.11

Referenced by:

§10.30(i), §10.65(ii)

Permalink:

http://dlmf.nist.gov/10.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

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10.31.2 $$K_0\left(z\right)=-\left(\ln \left(\frac{1}{2}z\right)+\gamma \right)I_0\left(z\right)+\frac{\frac{1}{4}{z}^2}{{(1!)}^2}+(1+\frac{1}{2})\frac{{(\frac{1}{4}{z}^2)}^2}{{(2!)}^2}+(1+\frac{1}{2}+\frac{1}{3})\frac{{(\frac{1}{4}{z}^2)}^3}{{(3!)}^2}+\mathrm{\cdots }.$$

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Symbols:

$\gamma$: Euler’s constant, $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.6.13

Referenced by:

§10.45

Permalink:

http://dlmf.nist.gov/10.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.31 and Ch.10

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10.31.3 $$I_{\nu }\left(z\right)I_{\mu }\left(z\right)={(\frac{1}{2}z)}^{\nu +\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\nu +\mu +k+1\right)}_k{(\frac{1}{4}{z}^2)}^k}{k!\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\mu +k+1\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.31

Permalink:

http://dlmf.nist.gov/10.31.E3

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §10.31 and Ch.10

14: 10.60 Sums

§10.60 Sums

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§10.60(i) Addition Theorems

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§10.60(ii) Duplication Formulas

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§10.60(iv) Compendia

… ►See also Watson (1944, Chapters 11 and 16).

15: 29 Lamé Functions

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16: 20.2 Definitions and Periodic Properties

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20.2.3 $${\theta }_3\left(z\right|\tau )={\theta }_3(z,q)=1+2\sum _{n=1}^{\mathrm{\infty }}{q}^{n^2}\cos \left(2nz\right),$$

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Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.3

Referenced by:

§20.10(i), §20.13, §20.14, §27.13(iv)

Permalink:

http://dlmf.nist.gov/20.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(i), §20.2 and Ch.20

… ►Corresponding expansions for ${\theta }_j^{\prime }\left(z\right|\tau )$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$. … ►For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$. … ►

20.2.8 $${\theta }_3\left(z+(m+n\tau )\pi \right|\tau )=q^{-n^2}{\mathrm{e}}^{-2\mathrm{i}nz}{\theta }_3\left(z\right|\tau ),$$

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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, $m$: integer, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.33.4 (in different notation)

Permalink:

http://dlmf.nist.gov/20.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(ii), §20.2 and Ch.20

… ►For $m,n\in \mathbb{Z}$, the $z$-zeros of ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, are $(m+n\tau )\pi$, $(m+\frac{1}{2}+n\tau )\pi$, $(m+\frac{1}{2}+(n+\frac{1}{2})\tau )\pi$, $(m+(n+\frac{1}{2})\tau )\pi$ respectively.

17: 10.19 Asymptotic Expansions for Large Order

… ►In these expansions $U_k(p)$ and $V_k(p)$ are the polynomials in $p$ of degree $3k$ defined in §10.41(ii). … ►with sectors of validity $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}\nu \le \frac{3}{2}\pi -\delta$. … ►

$$J_{\nu }^{\prime }\left(\nu +a{\nu }^{\frac{1}{3}}\right)\sim -\frac{2^{\frac{2}{3}}}{{\nu }^{\frac{2}{3}}}{\mathrm{Ai}}^{\prime }\left(-2^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{R_k(a)}{{\nu }^{2k/3}}+\frac{2^{\frac{1}{3}}}{{\nu }^{\frac{4}{3}}}\mathrm{Ai}\left(-2^{\frac{1}{3}}a\right)\sum _{k=0}^{\mathrm{\infty }}\frac{S_k(a)}{{\nu }^{2k/3}},$$ $|\mathrm{ph}\nu |\le \frac{1}{2}\pi -\delta$,

… ►with sectors of validity $-\frac{1}{2}\pi +\delta \le \mathrm{ph}\nu \le \frac{3}{2}\pi -\delta$ and $-\frac{3}{2}\pi +\delta \le \mathrm{ph}\nu \le \frac{1}{2}\pi -\delta$, respectively. … ►For higher coefficients in (10.19.8) in the case $a=0$ (that is, in the expansions of $J_{\nu }\left(\nu \right)$ and $Y_{\nu }\left(\nu \right)$), see Watson (1944, §8.21), Temme (1997), and Jentschura and Lötstedt (2012). …

18: 11.6 Asymptotic Expansions

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11.6.1 $${𝐊}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.29

Referenced by:

§11.6(i), §11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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$c_3(\lambda )=20{\lambda }^{-6}-4{\lambda }^{-4},$

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$c_4(\lambda )=70{\lambda }^{-8}-\frac{45}{2}{\lambda }^{-6}+\frac{3}{8}{\lambda }^{-4}.$

… ►See also Watson (1944, p. 336). … ►and for an estimate of the relative error in this approximation see Watson (1944, p. 336).

19: 22.8 Addition Theorems

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§22.8(i) Sum of Two Arguments

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§22.8(ii) Alternative Forms for Sum of Two Arguments

… ►A geometric interpretation of (22.8.20) analogous to that of (23.10.5) is given in Whittaker and Watson (1927, p. 530). … ►If sums/differences of the $z_j$’s are rational multiples of $K\left(k\right)$, then further relations follow. …is independent of $z_1$, $z_2$, $z_3$. …

20: 22 Jacobian Elliptic Functions

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