Abstract
In this paper, the hypercomplex Ruscheweyh derivative operator for special monogenic functions is defined. The representation in certain regions of such functions in terms of hypercomplex Ruscheweyh derivative bases of special monogenic polynomials (HRDBSMPs) are investigated. Precisely, we examine the approximation properties in different regions such as closed balls, open balls, closed regions surrounding closed balls, at the origin and for all entire special monogenic functions. Moreover, the order type and the T-rho-property for these bases are discussed. We also provide some interesting applications for some HRDBSMPs such as Bernoulli, Euler, and Bessel polynomials. The obtained results extend and enhance relevant results in the complex and Clifford setting.