# Watson sum

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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
###### §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: 10.44 Sums
###### §10.44(i) Multiplication Theorem
For results analogous to (10.23.7) and (10.23.8) see Watson (1944, §§11.3 and 11.41). …
##### 7: 11.10 Anger–Weber Functions
11.10.10 $S_{1}(\nu,z)=\sum_{k=0}^{\infty}\frac{(-1)^{k}(\tfrac{1}{2}z)^{2k}}{\Gamma% \left(k\!+\!\tfrac{1}{2}\nu+1\right)\Gamma\left(k\!-\!\tfrac{1}{2}\nu\!+\!1% \right)},$
11.10.30 ${\mathbf{J}_{\nu}\left(z\right)=}\\ {2\sin\left(\tfrac{1}{2}\nu\pi\right)\sum_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{% 2}\nu+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{k+\frac{1}{2}\nu+\frac{1}{2}}% \left(\tfrac{1}{2}z\right)}+{2\cos\left(\tfrac{1}{2}\nu\pi\right)\sideset{}{{}% ^{\prime}}{\sum}_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{2}\nu}\left(\tfrac{1}{2}z% \right)J_{k+\frac{1}{2}\nu}\left(\tfrac{1}{2}z\right)},$
11.10.31 ${\mathbf{E}_{\nu}\left(z\right)=}\\ {-2\cos\left(\tfrac{1}{2}\nu\pi\right)\sum_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}% {2}\nu+\frac{1}{2}}\left(\tfrac{1}{2}z\right)J_{k+\frac{1}{2}\nu+\frac{1}{2}}% \left(\tfrac{1}{2}z\right)}+{2\sin\left(\tfrac{1}{2}\nu\pi\right)\sideset{}{{}% ^{\prime}}{\sum}_{k=0}^{\infty}(-1)^{k}J_{k-\frac{1}{2}\nu}\left(\tfrac{1}{2}z% \right)J_{k+\frac{1}{2}\nu}\left(\tfrac{1}{2}z\right)},$
###### §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).
##### 8: 6.12 Asymptotic Expansions
6.12.5 $\mathrm{f}\left(z\right)=\frac{1}{z}\sum_{m=0}^{n-1}(-1)^{m}\frac{(2m)!}{z^{2m% }}+R_{n}^{(\mathrm{f})}(z),$
6.12.6 $\mathrm{g}\left(z\right)=\frac{1}{z^{2}}\sum_{m=0}^{n-1}(-1)^{m}\frac{(2m+1)!}% {z^{2m}}+R_{n}^{(\mathrm{g})}(z),$
##### 9: 20.8 Watson’s Expansions
###### §20.8 Watson’s Expansions
20.8.1 $\frac{\theta_{2}\left(0,q\right)\theta_{3}\left(z,q\right)\theta_{4}\left(z,q% \right)}{\theta_{2}\left(z,q\right)}=2\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q% ^{n^{2}}e^{i2nz}}{q^{-n}e^{-iz}+q^{n}e^{iz}}.$
See Watson (1935a). …
##### 10: Errata
• Subsection 17.7(iii)

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)”.

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

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.