Special Bi-Axial Monogenic Functions

In this paper we extend our recent work on axial monogenic functions in Rm+1 to functions which are monogenic in bi-axially symmetric domains of Rp+q. We show that an integral transform of a wide class of holomorphic functions of a single complex variable gives monogenic functions of this type. It is demonstrated that these integral transforms are related to plane wave monogenic functions. A bi-axial monogenic exponential function is defined using the exponential function of a complex variable and bounds are obtained on its modulus. Bi-axially symmetric monogenic generating functions are used to define generalisations of Gegenbauer polynomials and Hermite polynomials. Finally, bi-axial power functions are constructed using the above integral transform.