Reduced-quaternionic Mathieu functions, time-dependent Moisil-Teodorescu operators, and the imaginary-time wave equation

We construct a one-parameter family of generalized Mathieu functions, which are reduced quaternion-valued functions pair real variables lying in an ellipse, and we call λ-reduced quaternionic functions. prove that the λ-RQM kernel Moisil-Teodorescu operator D+λ (D is Dirac λ∈R∖{0}), form complete orthogonal system Hilbert space square-integrable λ-metamonogenic with respect to L2-norm over confocal ellipses. Further, introduce zero-boundary λ-RQM-functions, whose scalar part vanishes on boundary ellipse. The limiting values as eccentricity ellipse tends zero expressed terms Bessel first kind for unit disk. A connection between time-dependent solutions imaginary-time wave equation elliptical coordinate shown.