# conjugate

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##### 1: 18.19 Hahn Class: Definitions
18.19.1 $p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
18.19.2 $w(z;a,b,\overline{a},\overline{b})=\Gamma\left(a+iz\right)\Gamma\left(b+iz% \right)\Gamma\left(\overline{a}-iz\right)\Gamma\left(\overline{b}-iz\right),$
18.19.3 $w(x)=w(x;a,b,\overline{a},\overline{b})=|\Gamma\left(a+\mathrm{i}x\right)% \Gamma\left(b+\mathrm{i}x\right)|^{2},$
18.19.4 $h_{n}=\frac{2\pi\Gamma\left(n+a+\overline{a}\right)\Gamma\left(n+b+\overline{b% }\right)|\Gamma\left(n+a+\overline{b}\right)|^{2}}{\left(2n+2\Re\left(a+b% \right)-1\right)\Gamma\left(n+2\Re\left(a+b\right)-1\right)n!},$
##### 2: 18.20 Hahn Class: Explicit Representations
18.20.3 $w(x;a,b,\overline{a},\overline{b})p_{n}\left(x;a,b,\overline{a},\overline{b}% \right)=\frac{1}{n!}\delta_{x}^{n}\left(w(x;a+\tfrac{1}{2}n,b+\tfrac{1}{2}n,% \overline{a}+\tfrac{1}{2}n,\overline{b}+\tfrac{1}{2}n)\right).$
18.20.9 $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)=\frac{{\mathrm{i}}^{n}{\left% (a+\overline{a}\right)_{n}}{\left(a+\overline{b}\right)_{n}}}{n!}\*{{}_{3}F_{2% }}\left({-n,n+2\Re\left(a+b\right)-1,a+\mathrm{i}x\atop a+\overline{a},a+% \overline{b}};1\right).$
(For symmetry properties of $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)$ with respect to $a$, $b$, $\overline{a}$, $\overline{b}$ see Andrews et al. (1999, Corollary 3.3.4).) …
##### 3: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.4 $q_{n}(x)=\ifrac{p_{n}\left(x;a,b,\overline{a},\overline{b}\right)}{p_{n}\left(% \mathrm{i}a;a,b,\overline{a},\overline{b}\right)},$
18.22.13 $p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
$A(x)=(x+\mathrm{i}\overline{a})(x+\mathrm{i}\overline{b}),$
18.22.27 $\delta_{x}\left(p_{n}\left(x;a,b,\overline{a},\overline{b}\right)\right)=(n+2% \Re\left(a+b\right)-1)\*p_{n-1}\left(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline% {a}+\tfrac{1}{2},\overline{b}+\tfrac{1}{2}\right),$
18.22.28 $\delta_{x}\left(w(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline{a}+\tfrac{1}{2},% \overline{b}+\tfrac{1}{2})p_{n}(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline{a}+% \tfrac{1}{2},\overline{b}+\tfrac{1}{2})\right)=-(n+1)w(x;a,b,\overline{a},% \overline{b})p_{n+1}(x;a,b,\overline{a},\overline{b}).$
##### 4: 4.3 Graphics
Corresponding points share the same letters, with bars signifying complex conjugates. …
##### 6: 36.8 Convergent Series Expansions
36.8.5 $f_{n}(\zeta,\overline{\zeta})=c_{n}(\zeta)c_{n}(\overline{\zeta})\mathrm{Ai}% \left(\zeta\right)\mathrm{Bi}\left(\overline{\zeta}\right)+c_{n}(\zeta)d_{n}(% \overline{\zeta})\mathrm{Ai}\left(\zeta\right)\mathrm{Bi}'\left(\overline{% \zeta}\right)+d_{n}(\zeta)c_{n}(\overline{\zeta})\mathrm{Ai}'\left(\zeta\right% )\mathrm{Bi}\left(\overline{\zeta}\right)+d_{n}(\zeta)d_{n}(\overline{\zeta})% \mathrm{Ai}'\left(\zeta\right)\mathrm{Bi}'\left(\overline{\zeta}\right),$
##### 7: 10.34 Analytic Continuation
$I_{\nu}\left(\overline{z}\right)=\overline{I_{\nu}\left(z\right)},$
$K_{\nu}\left(\overline{z}\right)=\overline{K_{\nu}\left(z\right)}.$
For complex $\nu$ replace $\nu$ by $\overline{\nu}$ on the right-hand sides.
##### 9: 18.33 Polynomials Orthogonal on the Unit Circle
18.33.1 $\frac{1}{2\pi\mathrm{i}}\int_{|z|=1}\phi_{n}(z)\overline{\phi_{m}(z)}w(z)\frac% {\mathrm{d}z}{z}=\delta_{n,m},$
where the bar signifies complex conjugate. …
18.33.3 $\phi_{n}^{*}(z)=z^{n}\overline{\phi_{n}({\overline{z}}^{-1})}={\kappa_{n}}+% \sum_{\ell=1}^{n}\overline{\kappa}_{n,n-\ell}z^{\ell},$
where the bar again signifies compex conjugate. …
##### 10: 12.2 Differential Equations
Standard solutions are $U\left(a,\pm z\right)$, $V\left(a,\pm z\right)$, $\overline{U}\left(a,\pm x\right)$ (not complex conjugate), $U\left(-a,\pm iz\right)$ for (12.2.2); $W\left(a,\pm x\right)$ for (12.2.3); $D_{\nu}\left(\pm z\right)$ for (12.2.4), where …
###### §12.2(vi) Solution $\overline{U}\left(a,x\right)$; Modulus and Phase Functions
When $z$ $(=x)$ is real the solution $\overline{U}\left(a,x\right)$ is defined by …unless $a=\tfrac{1}{2},\tfrac{3}{2},\dots$, in which case $\overline{U}\left(a,x\right)$ is undefined. …Properties of $\overline{U}\left(a,x\right)$ follow immediately from those of $V\left(a,x\right)$ via (12.2.21). …