On an extension of harmonicity and holomorphy

The concept of harmonicity and holomorphy related to the Laplace equation (Formula Presented.) is extended with the use of (Formula Presented.), where Γ and Λ are C¹ -scalar functions of (Formula Presented.) for τ = 1, …, 5, respectively, t ∈ ℝ, a ∈ ℝ, and s_∗ is an arbitrary admissible function. We discuss the fundamental solutions for (0) (more precisely, of the corresponding linearized problem) which is a parabolic equation of the second kind. For effective solutions and τ ≡ 1, 2, 3, 4 (mod 8), it is convenient to involve the quaternionic structure, for τ ≡ 5, 6, 7, 0 (mod 8) - the paraquater-nionic structure. Physically, it is natural to describe with help of (0) relaxation processes attaching (x, y, z) to the first chosen parricle, (ξ, η, ζ) - to the second one, ¯τ to temperature, entropy or order parameter, and t - to time.