A linear ODE for the Omega function associated with the Euler function image and the Bernoulli function image Original Research Article

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