Bi-axial Gegenbauer functions of the first and second kind

Volume 37 / 1996

Alan Common Banach Center Publications 37 (1996), 181-187 DOI: 10.4064/-37-1-181-187


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 $


  • Alan Common

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