Let $S$ and $C$ denote the Fourier sine transform and the Fourier cosine transform, respectively, i.e., \begin{align*} S f(k) &= \sqrt{\frac2\pi} \, \int_0^\infty f(x) \sin(kx)\,dx, \\ C f(k) &= \sqrt{\frac2\pi} \, \int_0^\infty f(x) \cos(kx)\,dx. \end{align*} Recall that $S$ and $C$ are unitary selfadjoint operators on $L^2(\mathbb R_+)$.

Further let $\varphi_m(x) = e^{-x/2} L_m(x)$ for $m\in\mathbb N_0$, where $L_m$ is the $m$-th Laguerre polynomial. Then $\left\{\varphi_m\right\}_{m=0}^\infty$ forms an orthonormal basis of $L^2(\mathbb R_+)$. I am interested in a reference or a (short) proof of the following fact which is likely to be known:

$$ \left\langle C\varphi_m,S\varphi_n\right\rangle = \begin{cases} \displaystyle \frac2{\left(m+n+1\right)\pi} & \text{if $m\equiv n \pmod{2}$,} \\ \displaystyle \frac2{\left(m-n\right)\pi} & \text{if $m\not\equiv n \pmod{2}$.} \end{cases} $$

Here $\langle\;,\,\rangle$ is the scalar product on $L^2(\mathbb R_+)$.