Bi-axial Gegenbauer functions of the first and second kind

The classical orthogonal polynomials defined on intervals of the real line are related to many important branches of analysis and applied mathematics. Here a method is described to generalise this concept to polynomials defined on higher dimensional spaces using Bi-Axial Monogenic functions. The particular examples considered are Gegenbauer polynomials defined on the interval [-1,1] and the Gegenbauer functions of the second kind which are weighted Cauchy integral transforms over this interval of these polynomials. Related polynomials are defined which are orthogonal on the unit ball ${\displaystyle ^\left\{p\right\}\equiv \{\overrightarrow{x}\in {ℝ}^p;\left|\overrightarrow{x}\right|\le 1\}}$ using Bi-Axial Monogenic generating functions on ${\displaystyle {ℝ}^m}$. Then corresponding generalised Gegenbauer functions of the second kind are defined using generalised weighted Bi-Axial Monogenic Cauchy transforms of these polynomials over ${\displaystyle ^\left\{p\right\}}$. These generalised Gegenbauer functions of first and second kind reduce to the standard case when p=1 and are solutions of related second order differential equations which become identical in the one dimensional case.