# Legendre symbol

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## 1—10 of 41 matching pages

##### 1: 27.9 Quadratic Characters
For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. If $p$ divides $n$, then the value of $(n|p)$ is $0$. …The Legendre symbol $(n|p)$, as a function of $n$, is a Dirichlet character (mod $p$). …
27.9.2 $(2|p)=(-1)^{(p^{2}-1)/8}.$
27.9.3 $(p|q)(q|p)=(-1)^{(p-1)(q-1)/4}.$
##### 2: 27.1 Special Notation
 $d,k,m,n$ positive integers (unless otherwise indicated). … Legendre symbol; see §27.9.
##### 3: 19.15 Advantages of Symmetry
These reduction theorems, unknown in the Legendre theory, allow symbolic integration without imposing conditions on the parameters and the limits of integration (see §19.29(ii)). …
##### 4: 34.3 Basic Properties: $\mathit{3j}$ Symbol
###### §34.3(vii) Relations to Legendre Polynomials and Spherical Harmonics
34.3.19 $P_{l_{1}}\left(\cos\theta\right)P_{l_{2}}\left(\cos\theta\right)=\sum_{l}(2l+1% )\begin{pmatrix}l_{1}&l_{2}&l\\ 0&0&0\end{pmatrix}^{2}P_{l}\left(\cos\theta\right),$
34.3.21 $\int_{0}^{\pi}P_{l_{1}}\left(\cos\theta\right)P_{l_{2}}\left(\cos\theta\right)% P_{l_{3}}\left(\cos\theta\right)\sin\theta\,\mathrm{d}\theta=2\begin{pmatrix}l% _{1}&l_{2}&l_{3}\\ 0&0&0\end{pmatrix}^{2},$
##### 5: 14.6 Integer Order
14.6.5 ${\left(\nu+1\right)_{m}}\boldsymbol{Q}^{m}_{\nu}\left(x\right)=(-1)^{m}\left(x% ^{2}-1\right)^{m/2}\frac{{\mathrm{d}}^{m}\boldsymbol{Q}_{\nu}\left(x\right)}{{% \mathrm{d}x}^{m}}.$
##### 6: 19.12 Asymptotic Approximations
19.12.1 $K\left(k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left% (\tfrac{1}{2}\right)_{m}}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln\left(\frac{1}{k^{% \prime}}\right)+d(m)\right),$ $0<|k^{\prime}|<1$,
19.12.2 $E\left(k\right)=1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{3}{2}\right)_{m}}}{{\left(2\right)_{m}}m!}{k^{\prime% }}^{2m+2}\*\left(\ln\left(\frac{1}{k^{\prime}}\right)+d(m)-\frac{1}{(2m+1)(2m+% 2)}\right),$ $|k^{\prime}|<1$,
##### 7: Errata
• The Legendre polynomial $P_{n}$ was mistakenly identified as the associated Legendre function $P_{n}$ in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

• Equation (14.8.3)
14.8.3 $\mathsf{Q}_{\nu}\left(x\right)=\frac{1}{2}\ln\left(\frac{2}{1-x}\right)-\gamma% -\psi\left(\nu+1\right)+O\left(\left(1-x\right)\ln\left(1-x\right)\right),$ $\nu\neq-1,-2,-3,\dots$

The symbol $O\left(1-x\right)$ has been corrected to be $O\left(\left(1-x\right)\ln\left(1-x\right)\right)$.

Reported by Mark Ashbaugh on 2022-02-08

• Equation (14.8.9)
14.8.9 $\boldsymbol{Q}_{\nu}\left(x\right)=-\frac{\ln\left(x-1\right)}{2\Gamma\left(% \nu+1\right)}+\frac{\frac{1}{2}\ln 2-\gamma-\psi\left(\nu+1\right)}{\Gamma% \left(\nu+1\right)}+O\left(\left(x-1\right)\ln\left(x-1\right)\right),$ $\nu\neq-1,-2,-3,\dots$

The symbol $O\left(x-1\right)$ has been corrected to be $O\left(\left(x-1\right)\ln\left(x-1\right)\right)$.

Reported by Mark Ashbaugh on 2022-02-08

##### 10: 19.5 Maclaurin and Related Expansions
19.5.1 $K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),$
19.5.2 $E\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2% }F_{1}}\left({-\tfrac{1}{2},\tfrac{1}{2}\atop 1};k^{2}\right),$
19.5.4_1 $F\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{% \sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin}% ^{2}\phi\right),$
19.5.4_2 $E\left(\phi,k\right)=\sum_{m=0}^{\infty}\frac{{\left(-\tfrac{1}{2}\right)_{m}}% {\sin}^{2m+1}\phi}{(2m+1)m!}{{}_{2}F_{1}}\left({m+\tfrac{1}{2},\tfrac{1}{2}% \atop m+\tfrac{3}{2}};{\sin}^{2}{\phi}\right)k^{2m}=\sin\phi\,{F_{1}}\left(% \tfrac{1}{2};\tfrac{1}{2},-\tfrac{1}{2};\tfrac{3}{2};{\sin}^{2}\phi,k^{2}{\sin% }^{2}\phi\right),$
19.5.4_3 $\Pi\left(\phi,\alpha^{2},k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\sin}^{2m+1}\phi}{(2m+1)m!}{F_{1}}\left(m+\tfrac{1}{2};\tfrac{1}{% 2},1;m+\tfrac{3}{2};{\sin}^{2}\phi,\alpha^{2}{\sin}^{2}\phi\right)k^{2m},$