# normalizing constant

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##### 1: 33.13 Complex Variable and Parameters
33.13.1 $C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),$
##### 2: 30.1 Special Notation
where $d_{mn}(\gamma)$ is a normalization constant determined by …
##### 3: 31.9 Orthogonality
31.9.2 $\int_{\zeta}^{(1+,0+,1-,0-)}t^{\gamma-1}(1-t)^{\delta-1}(t-a)^{\epsilon-1}\*w_% {m}(t)w_{k}(t)\,\mathrm{d}t=\delta_{m,k}\theta_{m}.$
The normalization constant $\theta_{m}$ is given by
31.9.3 $\theta_{m}=(1-{\mathrm{e}}^{2\pi i\gamma})(1-{\mathrm{e}}^{2\pi i\delta})\zeta% ^{\gamma}(1-\zeta)^{\delta}(\zeta-a)^{\epsilon}\*\frac{f_{0}(q,\zeta)}{f_{1}(q% ,\zeta)}\left.\frac{\partial}{\partial q}\mathscr{W}\left\{f_{0}(q,\zeta),f_{1% }(q,\zeta)\right\}\right|_{q=q_{m}},$
For further information, including normalization constants, see Sleeman (1966a). …
##### 4: 33.5 Limiting Forms for Small $\rho$, Small $|\eta|$, or Large $\ell$
33.5.6 $C_{\ell}\left(0\right)=\frac{2^{\ell}\ell!}{(2\ell+1)!}=\frac{1}{(2\ell+1)!!}.$
33.5.9 $C_{\ell}\left(\eta\right)\sim\dfrac{e^{-\pi\eta/2}}{(2\ell+1)!!}\sim e^{-\pi% \eta/2}\dfrac{e^{\ell}}{\sqrt{2}(2\ell)^{\ell+1}}.$
##### 5: 33.7 Integral Representations
33.7.2 ${H^{-}_{\ell}}\left(\eta,\rho\right)=\frac{e^{-\mathrm{i}\rho}\rho^{-\ell}}{(2% \ell+1)!C_{\ell}\left(\eta\right)}\int_{0}^{\infty}e^{-t}t^{\ell-\mathrm{i}% \eta}(t+2\mathrm{i}\rho)^{\ell+\mathrm{i}\eta}\,\mathrm{d}t,$
33.7.3 ${H^{-}_{\ell}}\left(\eta,\rho\right)=\frac{-\mathrm{i}e^{-\pi\eta}\rho^{\ell+1% }}{(2\ell+1)!C_{\ell}\left(\eta\right)}\int_{0}^{\infty}\left(\frac{\exp\left(% -\mathrm{i}(\rho\tanh t-2\eta t)\right)}{(\cosh t)^{2\ell+2}}+\mathrm{i}(1+t^{% 2})^{\ell}\exp\left(-\rho t+2\eta\operatorname{arctan}t\right)\right)\,\mathrm% {d}t,$
33.7.4 ${H^{+}_{\ell}}\left(\eta,\rho\right)=\frac{\mathrm{i}e^{-\pi\eta}\rho^{\ell+1}% }{(2\ell+1)!C_{\ell}\left(\eta\right)}\int_{-1}^{-\mathrm{i}\infty}e^{-\mathrm% {i}\rho t}(1-t)^{\ell-\mathrm{i}\eta}(1+t)^{\ell+\mathrm{i}\eta}\,\mathrm{d}t.$
##### 6: 33.2 Definitions and Basic Properties
33.2.3 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)2^{-\ell-1}(\mp\mathrm% {i})^{\ell+1}M_{\pm\mathrm{i}\eta,\ell+\frac{1}{2}}\left(\pm 2\mathrm{i}\rho% \right),$
$F_{\ell}\left(\eta,\rho\right)$ is a real and analytic function of $\rho$ on the open interval $0<\rho<\infty$, and also an analytic function of $\eta$ when $-\infty<\eta<\infty$. The normalizing constant $C_{\ell}\left(\eta\right)$ is always positive, and has the alternative form
33.2.6 $C_{\ell}\left(\eta\right)=\dfrac{2^{\ell}\left((2\pi\eta/(e^{2\pi\eta}-1))% \prod_{k=1}^{\ell}(\eta^{2}+k^{2})\right)^{\ifrac{1}{2}}}{(2\ell+1)!}.$
##### 7: 33.9 Expansions in Series of Bessel Functions
33.9.3 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\frac{(2\ell+1)!}{(2% \eta)^{2\ell+1}}\rho^{-\ell}\*\sum_{k=2\ell+1}^{\infty}b_{k}t^{k/2}I_{k}\left(% \textstyle 2\sqrt{t}\right),$ $\eta>0$,
33.9.4 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\frac{(2\ell+1)!}{(2% \left|\eta\right|)^{2\ell+1}}\rho^{-\ell}\*\sum_{k=2\ell+1}^{\infty}\!\!b_{k}t% ^{k/2}J_{k}\left(\textstyle 2\sqrt{t}\right),$ $\eta<0$.
33.9.6 $G_{\ell}\left(\eta,\rho\right)\sim\frac{\rho^{-\ell}}{(\ell+\frac{1}{2})% \lambda_{\ell}(\eta)C_{\ell}\left(\eta\right)}\*\sum_{k=2\ell+1}^{\infty}(-1)^% {k}b_{k}t^{k/2}K_{k}\left(\textstyle 2\sqrt{t}\right),$
##### 8: 33.6 Power-Series Expansions in $\rho$
33.6.1 $F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}A_{k}^{\ell}(\eta)\rho^{k},$
33.6.2 $F_{\ell}'\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\sum_{k=\ell+1}^{% \infty}kA_{k}^{\ell}(\eta)\rho^{k-1},$
##### 10: 33.16 Connection Formulas
33.16.1 $F_{\ell}\left(\eta,\rho\right)=\dfrac{(2\ell+1)!C_{\ell}\left(\eta\right)}{(-2% \eta)^{\ell+1}}f\left(1/\eta^{2},\ell;-\eta\rho\right),$
33.16.2 $G_{\ell}\left(\eta,\rho\right)=\dfrac{\pi(-2\eta)^{\ell}}{(2\ell+1)!C_{\ell}% \left(\eta\right)}h\left(1/\eta^{2},\ell;-\eta\rho\right),$