A bicomplex proportional fractional $(\vartheta,\varphi)-$weighted Cauchy-Riemann operator using Riemann-Liouville derivatives with respect to an hyperbolic-valued function

TL;DR

This paper introduces a new bicomplex proportional fractional Cauchy-Riemann operator based on Riemann-Liouville derivatives with hyperbolic-valued functions, along with its fractional Borel-Pompeiu formula.

Contribution

It defines the first such operator involving hyperbolic-valued functions and proves its associated fractional Borel-Pompeiu formula.

Findings

Introduction of a novel bicomplex proportional fractional operator

Proof of the fractional Borel-Pompeiu formula for the operator

Establishment of the operator's foundational properties

Abstract

Based on the Riemann-Liouville derivatives with respect to functions taking values in the set of hyperbolic numbers, we consider a novel bicomplex proportional fractional $(\vartheta,\varphi)-$weighted Cauchy-Riemann operator, involving weights hyperbolic orthogonal bicomplex functions. This operator is defined for the first time here, and its associated fractional Borel-Pompeiu formula is proved as the main result.