Watson sum

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1: 16.4 Argument Unity

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Watson’s Sum

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2: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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Gasper–Rahman $q$-Analog of Watson’s ${}_3{}^{}F_2^{}$ Sum

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Andrews’ $q$-Analog of the Terminating Version of Watson’s ${}_3{}^{}F_2^{}$ Sum (16.4.6)

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3: 10.23 Sums

§10.23 Sums

… ► … ►see Watson (1944, §16.13), and for further generalizations see Watson (1944, Chapter 16) and Erdélyi et al. (1953b, §7.10.1). … ►This result is proved in Watson (1944, Chapter 18) and further information is provided in this reference, including the behavior of the series near $x=0$ and $x=1$. … ►For other types of expansions of arbitrary functions in series of Bessel functions, see Watson (1944, Chapters 17–19) and Erdélyi et al. (1953b, §§ 7.10.2–7.10.4). …

4: 22.6 Elementary Identities

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§22.6(i) Sums of Squares

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5: 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. …

6: 11.10 Anger–Weber Functions

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11.10.10 $$S_1(\nu ,z)=\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(\frac{1}{2}z)}^{2k}}{\mathrm{\Gamma }\left(k+\frac{1}{2}\nu +1\right)\mathrm{\Gamma }\left(k-\frac{1}{2}\nu +1\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Referenced by:

(11.11.5), (11.11.6)

Permalink:

http://dlmf.nist.gov/11.10.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(iii), §11.10 and Ch.11

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11.10.30 $${𝐉}_{\nu }\left(z\right)=2\sin \left(\frac{1}{2}\nu \pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{J}_{k-\frac{1}{2}\nu +\frac{1}{2}}\left(\frac{1}{2}z\right)J_{k+\frac{1}{2}\nu +\frac{1}{2}}\left(\frac{1}{2}z\right)+2\cos \left(\frac{1}{2}\nu \pi \right)\underset{k=0\phantom{\prime }}{\overset{\mathrm{\infty }\phantom{\prime }}{{\sum }^{\prime }}}{(-1)}^k{J}_{k-\frac{1}{2}\nu }\left(\frac{1}{2}z\right)J_{k+\frac{1}{2}\nu }\left(\frac{1}{2}z\right),$$

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/11.10.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(viii), §11.10 and Ch.11

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11.10.31 $${𝐄}_{\nu }\left(z\right)=-2\cos \left(\frac{1}{2}\nu \pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{J}_{k-\frac{1}{2}\nu +\frac{1}{2}}\left(\frac{1}{2}z\right)J_{k+\frac{1}{2}\nu +\frac{1}{2}}\left(\frac{1}{2}z\right)+2\sin \left(\frac{1}{2}\nu \pi \right)\underset{k=0\phantom{\prime }}{\overset{\mathrm{\infty }\phantom{\prime }}{{\sum }^{\prime }}}{(-1)}^k{J}_{k-\frac{1}{2}\nu }\left(\frac{1}{2}z\right)J_{k+\frac{1}{2}\nu }\left(\frac{1}{2}z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/11.10.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(viii), §11.10 and Ch.11

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§11.10(x) Integrals and Sums

… ►For sums see Hansen (1975, pp. 456–457) and Prudnikov et al. (1990, §§6.4.2–6.4.3).

7: 6.12 Asymptotic Expansions

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6.12.5 $$\mathrm{f}\left(z\right)=\frac{1}{z}\sum _{m=0}^{n-1}{(-1)}^m\frac{(2m)!}{z^{2m}}+R_n^{(\mathrm{f})}(z),$$

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Symbols:

$!$: factorial (as in $n!$), $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{f})}(z)$: remainder term

Referenced by:

§6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

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6.12.6 $$\mathrm{g}\left(z\right)=\frac{1}{z^2}\sum _{m=0}^{n-1}{(-1)}^m\frac{(2m+1)!}{z^{2m}}+R_n^{(\mathrm{g})}(z),$$

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Symbols:

$!$: factorial (as in $n!$), $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{g})}(z)$: remainder term

Permalink:

http://dlmf.nist.gov/6.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

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8: 10.44 Sums

§10.44 Sums

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§10.44(i) Multiplication Theorem

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§10.44(ii) Addition Theorems

… ►For results analogous to (10.23.7) and (10.23.8) see Watson (1944, §§11.3 and 11.41). … ►

§10.44(iv) Compendia

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9: 20.8 Watson’s Expansions

§20.8 Watson’s Expansions

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20.8.1 $$\frac{{\theta }_2(0,q){\theta }_3(z,q){\theta }_4(z,q)}{{\theta }_2(z,q)}=2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n^2}{\mathrm{e}}^{\mathrm{i}2nz}}{q^{-n}{\mathrm{e}}^{-\mathrm{i}z}+q^n{\mathrm{e}}^{\mathrm{i}z}}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.8 and Ch.20

►See Watson (1935a). …

10: Errata

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The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog of (17.7.8)” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Watson’s ${}_3{}^{}F_2^{}$ Sum (16.4.6)”. The title of the paragraph which was previously “Andrews’ Terminating $q$-Analog” has been changed to “Andrews’ $q$-Analog of the Terminating Version of Whipple’s ${}_3{}^{}F_2^{}$ Sum (16.4.7)”.

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26.7.6 $$B\left(n+1\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B\left(k\right)$$

Originally this equation appeared with $B\left(n\right)$ in the summation, instead of $B\left(k\right)$.

Reported 2010-11-07 by Layne Watson.

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