Bi-axial Gegenbauer Functions of the Second Kind

Bi-axially symmetric monogenic generating functions on Rp + q have been used recently to define generalisations of Gegenbauer polynomials. These polynomials are orthogonal on the unit ball in Rp. Generalised Cauchy transforms of these polynomials are used to define corresponding bi-axial Gegenbauer functions of the second kind. It is demonstrated that these functions of the second kind satisfy second order differential equations related to those satisfied by the corresponding bi-axial Gegenbauer polynomials.