The product formula and convolution structure associated with the generalized Hermite polynomials (Q1801559)

Generalized Hermite polynomials are orthogonal with respect to \(| x|^{2a}\exp(-| x|^ 2)\) on \((-\infty,\infty)\). These functions can be represented as Laguerre polynomials. The product formula for Laguerre polynomials gives a product formula for generalized Hermite polynomials. For \(a\geq 1/2\), this leads to a bounded convolution structure. For \(0\leq a<1/2\) the kernel is bounded in weighted spaces. In particular, the classical Hermite polynomials did not have a known product formula or convolution formula before this paper.