A linear ODE for the Omega function associated with the Euler function Eα(z) and the Bernoulli function Bα(z)

The authors derive a linear ODE (ordinary differential equation) whose particular solution is the Butzer-Flocke-Hauss complete real-parameter Omega function Ω(w), which is associated with the complex-index Bernoulli function Bα(z) and with the complex-index Euler function Eα(z). This is accomplished here with the aid of an integral representation of the alternating Mathieu series S˜(w). A new integral representation and some two-sided bounding inequalities are also given for the Omega function.