Boundary Value Problems Of Generalized Poly-analytic Functions With Shift And Conjugation

In this paper,we study the Riemann-Hilbert problem for a class of generalized polyanalytic functions with Carleman shift and conjugation on the Lyapunov curve,which are null-solutions of a class of iteratived Beltrami equations with order n(for short,generalized ? polyanalytic functions of order n).The first innovation of this paper is in the third chapter,associating with the Cauchy-Pompeiu formula for generalized ? analytic functions,a weakly singular kernel is constructed by introducing homeomorphism transformation between the Lyapunov curves,and thus a new integral representation of generalized ? polyanalytic functions is also obtained by the Fredholm integral equation method,which develop and extend those results of Kat and Katz from Russian.The second innovation of this paper is in the fourth chapter,we introduce the concept of the canonical matrix and develop its theory in the context of some Beltrami equations.Appling the construction method,the explicit canonical factorization of triangular matrix functions and the estimate of its partial indices are obtained respectively,which extend some theory of matrix functions factorization established by G.S.Litvinchuk et al.The third innovation of this paper is in Chapter 5,by combing the Chapter 4 with the decomposition theorem of higher-order ?-analytic functions,we discuss the Riemann-Hilbert problem with Carleman shift and conjugation,and obtain its general solution and the solvable conditions of the problem under consideration.These results extend boundary value problems(including Hilbert and Haseman problem)for polyanalytic functions,also develop the theory of classical Carleman type problem.Finally,in Chapter 6,we summarize the main content of this paper,and explain the extended open problems and exisiting difficulties,On this basis,we also make some constructive predictions and marks for the Riemann-Hilber problem with Carleman translation and conjugation.