Groups of partial differential operators and the generalized Bessel functions (Q1937755)

The generalize Bessel function (GBF) \[ U^{(m)}_{\nu}(z)=U_{\nu}(z,m)=\sum_{k=0}^{\infty}\frac{(z/m)^{\nu +mk}}{k!\Gamma[(m-1)k+\nu+1]}, \] where \(m\geq 2\) is integer, \(\nu=\nu_m=-(m-1)p\), \(\Gamma(t)\) is the Euler gamma-function, \(p\) is a complex parameter, and \(z\) is a complex variable, solves the differential equation \[ \begin{multlined} \left[(m-1)\frac{d}{dz}+\frac{\nu+1}{z}\right]\left[(m-1)\frac{d}{dz}+\frac{\nu+2}{z}\right]\dotsm\\ \left[(m-1)\frac{d}{dz}+\frac{\nu+m-1}{z}\right]\times \left(\frac{d}{dz}-\frac{\nu}{z}\right) U_{\nu}(z,m)=U_{\nu}(z,m).\end{multlined} \] In the paper, some addition theorems and generating functions for GBF are derived by means of some algebra constructed for a group of first-order partial differential operators based on the recurrence relations for these functions.