Fourier sine and cosine transforms and Laguerre polynomials

Question

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 $L^2(\mathbb R_+)$.

Answer 1

Here is a derivation starting from the Fourier transforms given in Orthogonal polynomials on the unit circle associated with the Laguerre polynomials. The Fourier cosine and sine transforms of $\phi_m$ are equal to the real and imaginary parts of $\sqrt{2/\pi}(1-z)z^m$, with $z=(2k-i)/(2k+i)$ on the unit circle in the complex plane. Notice that $idz/dk=(1-z)^2$ and that $z$ varies over the lower half of the unit circle when $k$ varies over the positive real axis. The desired integral is $$\langle C\phi_m,S\phi_n\rangle=\frac{2}{\pi}\int_{0}^{\infty}[{\rm Re}\,(1-z)z^m][{\rm Im}\, (1-z)z^{n}]dk$$ $$\qquad = \frac{2i}{\pi}\int_{|z|=1,{\rm Im}\,z<0}[{\rm Re}\,(1-z)z^m][{\rm Im}\, (1-z)z^{n}]\frac{dz}{(1-z)^2}$$ $$\qquad = -\frac{2}{\pi}\int_{-\pi}^0[{\rm Re}\,(1-e^{i\phi})e^{im\phi}][{\rm Im}\, (1-e^{i\phi})e^{in\phi}]\frac{e^{i\phi}}{(1-e^{i\phi})^2}d\phi$$ $$\qquad=\frac{(2 m+1) +(-1)^{m+n+1}(2 n+1) }{(m-n) (m+n+1)\pi}$$ $$\qquad = \begin{cases} \displaystyle \frac2{\left(m+n+1\right)\pi} & \text{if $m+n$ even,} \\ \displaystyle \frac2{\left(m-n\right)\pi} & \text{if $m+n$ odd.} \end{cases}$$