On a convolution property characterizing the Laguerre functions (Q750685)

Consider the Laguerre functions \[ \ell^ p_ n(t)=(-1)^ n\sqrt{2p}L_ n(2pt)e^{-pt}\text{ (with parameter } p>0), \] where the \(L_ n\) are the Laguerre polynomials with parameter \(\alpha =0\). \(\{\ell^ p_ n(t)\}^{\infty}_{n=0}\) forms a complete orthonormal system in \(L^ 2([0,\infty))\). A well known and often used property of the Laguerre functions is the convolution property: \[ \sqrt{2p}\ell^ p_ i*\ell^ p_ j=\ell^ p_{i+j}+\ell^ p_{i+j+1}\text{ for all } i,j\geq 0. \] It is the objective to prove that the system of Laguerre functions is the only complete and orthonormal system of \(L^ 2([0,\infty))\) satisfying the convolution property.