A note on incomplete integrals of cylindrical functions (Q1824741)

Representations for integrals of the type \[ C_{\mu,\nu}(a,z)=\int^{z}_{0}e^{at}t^{\mu}C_{\nu}(t)dt, \] where \(C_{\nu}(t)\) is a cylindrical function (i.e., a Bessel or Hankel function), are given in closed form by using Kampé de Fériet double hypergeometric functions. Similar integrals are considered with the exponential function replaced with a sine or cosine function. Reduction formulae for the Kampé de Fériet functions associated with these integrals are provided for some cases.