Ultraspherical expansions and pseudo analytic functions

Abstract

This paper takes up the function theoretic approach to the study of ultraspherica l expansions, their conjugates, the associated elliptic equations, and first order systems. The theory of pseudo analytic functions and BergmanGilbert type integral operators are employed, and the relation between these two approaches is examined. Throughout, results obtained are analogs of well known theorems from the theory of analytic functions of a single complex variable, and the related study of harmonic functions and Fourier series. The study of trigonometric series, analytic functions, Laplace's equation, and the Cauchy-Riemann system are all in a sense equivalent. Since this study has proven to be one of the most fruitful in mathematics, and since Laplace's equation is just one specific elliptic partial differential equation, analogous developments should be expected for more general elliptic equations. In particular, it is natural to hope for a relationship with analytic functions corresponding to that found in the case of harmonic functions, u = ΈLe(f)9 which has proved so useful in the study of Laplace's equation and expansions in the associated special functions (trigonometric series). For u the solution of an elliptic equation more general than Laplace's, two approaches are apparent: (1) Generalize the operation of taking the real part. That