root datum of a Riemannian symmetric space, $\mathbb{D}$ is the algebra of radial components along $A$ of $G$-invariant differential operators on $G/K$. The analogue of the radial component of the Laplace–Beltrami operator is the operator $D_{p_L}$, where $p_L\in S({\mathfrak{a}}_{\mathbb{C}})^W$ is the polynomial $p_L(\lambda)=\langle\lambda,\lambda\rangle$. Then $D_{p_L}=L+\langle\rho,\rho\rangle$ where $$L=L_{\mathfrak{a}}+\sum_{\alpha\in\Sigma^+}m_\alpha\coth \alpha \partial(x_\alpha);$$ here $L_{\mathfrak{a}}$ is the Euclidean Laplace operator on ${\mathfrak{a}}$, and $\coth \alpha=\frac{1+e^{-2\alpha}}{1-e^{-2\alpha}}$.

Let $\lambda\in{\mathfrak{a}}_{\mathbb{C}}^*$ be fixed. The *hypergeometric function with spectral parameter* $\lambda\in{\mathfrak{a}}_{\mathbb{C}}^*$ is the unique analytic $W$-invariant function $\varphi_\lambda$ on ${\mathfrak{a}}$ that satisfies the system of differential equations $$D_p\varphi = p(\lambda)\varphi,\quad p\in S({\mathfrak{a}}_{\mathbb{C}})^W$$ and is normalized by $\varphi_\lambda(0)=1$.

In the case $n=1$, $\Sigma^+$ consists of at most two elements, $\alpha$ and $2\alpha$. Identify ${\mathfrak{a}}$ and ${\mathfrak{a}}^*$ with ${\mathbb{R}}$ by setting $x_\alpha/2\equiv 1$ and $\alpha\equiv 1$. Then ${\mathfrak{a}}_+\simeq (0,\infty)$, and $W=\{-1,1\}$ acts on ${\mathbb{R}}$ and ${\mathbb{C}}$ by multiplication. The algebra $\mathbb{D}$ is generated by a single element, for example the operator $D_{\rho_L}=L+\rho^2$, where $\rho=m_\alpha/2+m_{2\alpha}$. The hypergeometric system of differential equations used to define $\varphi$ reduces to the sdifferential equation $$\frac{d^2\varphi}{dz^2}+(m_\alpha\coth z+m_{2\alpha}\coth(2z))\frac{d\varphi}{dz}=(\lambda^2-\rho^2)\varphi$$ which may be transformed into the hypergeometric differential equation $$\zeta(1-\zeta)\frac{d^2\psi}{d\zeta^2}+(c-(1+a+b)\zeta)\frac{d\psi}{d\zeta}-ab\zeta=0$$ where $\zeta=\frac{1}{2}(1-\cosh z)$, $a=\frac{\lambda+\rho}{2}$, $b=\frac{-\lambda+\rho}{2}$, and $c=\frac{m_\alpha+m_{2\alpha}+1}{2}$. The solution $\varphi_\lambda$ is therefore the Jacobi functions $$\varphi_\lambda(t)={_2}F_1\Bigl(\frac{m_\alpha/2+m_{2\alpha}+\lambda}{2},\frac{m_\alpha/2+m_{2\alpha}-\lambda}{2};\frac{m_\alpha+m_{2\alpha}+1}{2};-\sinh^2t\Bigr)$$ which are well known 