We denote $\alpha > -1/2$, $E$ the $\mathbb{R}$-vector space of functions of class $\mathcal{C}^\infty$ on $[-1,1]$ with real values, $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ and $$S_\alpha(f,g) = \int_{-1}^{1} f(t)g(t)\left(1-t^2\right)^{\alpha - \frac{1}{2}} \mathrm{~d}t$$ Show that $$\forall (f,g) \in E^2, \quad S_\alpha\left(\varphi_\alpha(f), g\right) = S_\alpha\left(f, \varphi_\alpha(g)\right)$$ One may calculate the derivative of $t \mapsto \left(1-t^2\right)^{\alpha+\frac{1}{2}} f'(t)$.
We denote $\alpha > -1/2$, $E$ the $\mathbb{R}$-vector space of functions of class $\mathcal{C}^\infty$ on $[-1,1]$ with real values,
$$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$
and
$$S_\alpha(f,g) = \int_{-1}^{1} f(t)g(t)\left(1-t^2\right)^{\alpha - \frac{1}{2}} \mathrm{~d}t$$
Show that
$$\forall (f,g) \in E^2, \quad S_\alpha\left(\varphi_\alpha(f), g\right) = S_\alpha\left(f, \varphi_\alpha(g)\right)$$
One may calculate the derivative of $t \mapsto \left(1-t^2\right)^{\alpha+\frac{1}{2}} f'(t)$.