analytic continuation in n

by Nabiha ben abdallah   Last Updated January 16, 2018 17:20 PM

how can I pass from a variable function a natural integer n to a function that depends on a complex number by analytic continuation in n

The Jacobi function $\varphi_\mu ^{(\alpha,\beta)}(t),\ \mu \in \mathbb{C} ,$ is defined by $\forall t\in\mathbb{R},\quad\varphi_{\mu}^{(\alpha,\beta)}(t)= \,_2F_1\left(\frac{\rho+i\mu}{2},\frac{\rho-i\mu}{2};\alpha+1;-(\sinh t)^2\right),$ where $_2F_1$ is the Gauss hypergeometric function. $R_n^{(\alpha,\beta)}(t)= \,_2F_1\left(-n;n + \rho;\alpha+1;\frac{1}{2} (1-t)\right),$

if $n=\frac{i\mu-\rho}{2}$, $\forall\mu\in\mathbb{C},\ \forall t \in\mathbb{R},\quad\varphi_{\mu}^{(\alpha,\beta)}(t)= R_{\frac{i\mu-\rho}{2}}^{(\alpha,\beta)}(\cosh(2t))$

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