Laguerre entire functions and Sturm-Liouville hypergroups (Q2777864)

The author studies the hypergroup \(H_\theta\) on \(\mathbb R_+\) corresponding to the differential operator NEWLINE\[NEWLINE (\Delta_\theta u)(x)=xu''(x)+(\theta +1)u'(x),\quad \theta >-1/2. NEWLINE\]NEWLINE The generalized characters of \(H_\theta\) are expressed via the Bessel function \(J_\theta\). The convolution in \(H_\theta\) preserves the class of all bounded Radon measures whose generalized Laplace transforms (defined by the generalized characters) are entire functions obtained as limits of sequences of polynomials with real non-positive zeroes uniform on compact subsets of \(\mathbb C\). The hypergroup \(H_\theta\) (connected also with the Bessel-Kingman hypergroup) is used for constructing a biorthogonal Appell basis in \(L_2(\mathbb R^+,d\gamma_\theta)\), where \(d\gamma_\theta =[\Gamma (\theta +1)]^{-1}x^\theta e^{-x} dx\), and the corresponding spaces of test functions and distributions.