Jacobi symbol

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1: 27.9 Quadratic Characters
27.9.3 $(p|q)(q|p)=(-1)^{(p-1)(q-1)/4}.$
If an odd integer $P$ has prime factorization $P=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)=\prod_{r=1}^{\nu\left(n\right)}{(n|p_{r})}^{a_{r}}$, with $(n|1)=1$. The Jacobi symbol $(n|P)$ is a Dirichlet character (mod $P$). …
2: 27.1 Special Notation
 $d,k,m,n$ positive integers (unless otherwise indicated). … Jacobi symbol; see §27.9. …
3: 20.1 Special Notation
Primes on the $\theta$ symbols indicate derivatives with respect to the argument of the $\theta$ function. …
4: 18.30 Associated OP’s
18.30.5 $\frac{(-1)^{n}{\left(\alpha+\beta+c+1\right)_{n}}n!\,P^{(\alpha,\beta)}_{n}% \left(x;c\right)}{{\left(\alpha+\beta+2c+1\right)_{n}}{\left(\beta+c+1\right)_% {n}}}=\sum_{\ell=0}^{n}\frac{{\left(-n\right)_{\ell}}{\left(n+\alpha+\beta+2c+% 1\right)_{\ell}}}{{\left(c+1\right)_{\ell}}{\left(\beta+c+1\right)_{\ell}}}% \left(\tfrac{1}{2}x+\tfrac{1}{2}\right)^{\ell}\*{{}_{4}F_{3}}\left({\ell-n,n+% \ell+\alpha+\beta+2c+1,\beta+c,c\atop\beta+\ell+c+1,\ell+c+1,\alpha+\beta+2c};% 1\right),$
5: 18.12 Generating Functions
18.12.2_5 ${{}_{2}F_{1}}\left({\gamma,\alpha+\beta+1-\gamma\atop\alpha+1};\frac{1-R-z}{2}% \right)\*{{}_{2}F_{1}}\left({\gamma,\alpha+\beta+1-\gamma\atop\beta+1};\frac{1% -R+z}{2}\right)=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}{\left(% \alpha+\beta+1-\gamma\right)_{n}}}{{\left(\alpha+1\right)_{n}}{\left(\beta+1% \right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$ $R=\sqrt{1-2xz+z^{2}}$, $|z|<1$,
18.12.3 $(1+z)^{-\alpha-\beta-1}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}(\alpha+\beta+1),% \tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2(x+1)z}{(1+z)^{2}}\right)=% \sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right)_{n}}}{{\left(\beta+1% \right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$ $|z|<1$,
18.12.3_5 $\frac{1+z}{(1-2xz+z^{2})^{\beta+\frac{3}{2}}}=\sum_{n=0}^{\infty}\frac{{\left(% 2\beta+2\right)_{n}}}{{\left(\beta+1\right)_{n}}}P^{(\beta+1,\beta)}_{n}\left(% x\right)z^{n},$ $|z|<1$,
18.12.4 $(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{(\lambda)}_{n}\left(x\right)z^{% n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\tfrac% {1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x% \right)z^{n},$ $|z|<1$.
6: 18.7 Interrelations and Limit Relations
18.7.1 $C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{{\left(% \lambda+\frac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_% {n}\left(x\right),$
18.7.2 $P^{(\alpha,\alpha)}_{n}\left(x\right)=\frac{{\left(\alpha+1\right)_{n}}}{{% \left(2\alpha+1\right)_{n}}}C^{(\alpha+\frac{1}{2})}_{n}\left(x\right).$
18.7.15 $C^{(\lambda)}_{2n}\left(x\right)=\frac{{\left(\lambda\right)_{n}}}{{\left(% \tfrac{1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},-\frac{1}{2})}_{n}\left(2x^{2% }-1\right),$
18.7.16 $C^{(\lambda)}_{2n+1}\left(x\right)=\frac{{\left(\lambda\right)_{n+1}}}{{\left(% \frac{1}{2}\right)_{n+1}}}xP^{(\lambda-\frac{1}{2},\frac{1}{2})}_{n}\left(2x^{% 2}-1\right).$
7: 18.1 Notation
18.1.1 $C^{(0)}_{n}\left(x\right)=\frac{2}{n}T_{n}\left(x\right)=\frac{2(n-1)!}{{\left% (\tfrac{1}{2}\right)_{n}}}P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(x\right),$ $n=1,2,3,\dots$.
8: 18.14 Inequalities
18.14.1 $|P^{(\alpha,\beta)}_{n}\left(x\right)|\leq P^{(\alpha,\beta)}_{n}\left(1\right% )=\frac{{\left(\alpha+1\right)_{n}}}{n!},$ $-1\leq x\leq 1$, $\alpha\geq\beta>-1$, $\alpha\geq-\tfrac{1}{2}$,
18.14.2 $|P^{(\alpha,\beta)}_{n}\left(x\right)|\leq|P^{(\alpha,\beta)}_{n}\left(-1% \right)|=\frac{{\left(\beta+1\right)_{n}}}{n!},$ $-1\leq x\leq 1$, $\beta\geq\alpha>-1$, $\beta\geq-\tfrac{1}{2}$.
18.14.3_5 $\left(\tfrac{1}{2}(1+x)\right)^{\beta/2}\left|P^{(\alpha,\beta)}_{n}\left(x% \right)\right|\leq P^{(\alpha,\beta)}_{n}\left(1\right)=\frac{{\left(\alpha+1% \right)_{n}}}{n!},$ $-1\leq x\leq 1$, $\alpha,\beta\geq 0$.
18.14.25 $\sum_{m=0}^{n}\frac{{\left(\lambda+1\right)_{n-m}}}{(n-m)!}\frac{{\left(% \lambda+1\right)_{m}}}{m!}\mbox{\displaystyle\frac{P^{(\alpha,\beta)}_{m}% \left(x\right)}{P^{(\beta,\alpha)}_{m}\left(1\right)}\geq 0},$ $x\geq-1$, $\alpha+\beta\geq\lambda\geq 0$, $\beta\geq-\tfrac{1}{2}$, $n=0,1,\dots$.
9: 18.11 Relations to Other Functions
18.11.1 $\mathsf{P}^{m}_{n}\left(x\right)={\left(\tfrac{1}{2}\right)_{m}}(-2)^{m}(1-x^{% 2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x\right)={\left(n+1\right)_{m% }}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}P^{(m,m)}_{n-m}\left(x\right),$ $0\leq m\leq n$.
10: 18.5 Explicit Representations
18.5.7 $P^{(\alpha,\beta)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(n+\alpha+% \beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{n-\ell}}}{\ell!\;(n-\ell)!}% \left(\frac{x-1}{2}\right)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{2% }F_{1}}\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right),$
18.5.8 $P^{(\alpha,\beta)}_{n}\left(x\right)=2^{-n}\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0% pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+% 1)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}\left(\frac{x+1}{2}\right)^{n}% {{}_{2}F_{1}}\left({-n,-n-\beta\atop\alpha+1};\frac{x-1}{x+1}\right),$