On a left-α-ψ-hyperholomorphic Bergman space

Given a structural set (Formula presented.), some papers [M. Shapiro, V. Vasilevski, (Complex Var Theory Appl. 1995;27:17–46 and 1995;27:67–96)] study the quaternionic ψ-Fueter operator calculus, where the Fueter-type operator is (Formula presented.) The aim of this paper is to present the function theory and the Bergman theory induced by the ψ-Fueter-type operator: (Formula presented.) where a is a quaternionic constant and f is a continuously differentiable function. Particularly, this work shows the Stokes and Borel–Pompieu formulas associated to (Formula presented.) and presents several results of the Bergman theory such as the existence of the Bergman kernel, Bergman projection and their conformal covariant properties.