  The main goal of this paper is to construct a proportional analogues of the
quaternionic fractional Fueter-type operator recently introduced in the
literature. We start by establishing a quaternionic version of the well-known
proportional fractional integral and derivative with respect to a real-valued
function via the Riemann-Liouville fractional derivative. As a main result, we
prove a quaternionic proportional fractional Borel-Pompeiu formula based on a
quaternionic proportional fractional Stokes formula. This tool in hand allows
us to present a Cauchy integral type formula for the introduced quaternionic
proportional fractional Fueter-type operator with respect to a real-valued
function.
