Bases of special functions and their domains of convergence
A systematic approach to expansions of analytic functions in series of special functions is presented. Many expansions of this kind are identified with eigenfunction expansions for differential operators in the complex domain. Central ponits of our theory are the construction of biorthogonal canonical systems of eigen - and associated functions and the determination of the domains of convergence of the corresponding eigenfunction expansions.
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Foundations of the theory
First order differential systems with a regular singular point
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2x2 differential system 5log associated functions associated vectors assume Bessel bilinear form biorthogonal canonical systems circle compact subsets completes the proof converges absolutely uniformly CSRF defined according differential operator domains of convergence easily verifies eigenfunction expansions eigenfunctions and associated eigenvectors and associated exists a unique expansions in series Floquet eigenvalue problem form a fundamental formal adjoint Frechet spaces function f functions of mathematical fundamental system Furthermore generalised Fourier coefficients given holomorphic function holomorphic with respect hypergeometric functions immediate consequence isolated point Let us denote m-fold products mapped biholomorphically mathematical physics matrix matrix-valued function natural number obtain open subset order 2x2 differential order differential equation pair of Floquet radius regular singular point Remark ring-shaped region root function second order differential section 4.1 series converges absolutely simply connected Slog special functions suitable domains Suppose terms of eigenfunctions uniformly on compact unique expansion uniquely determined Whittaker functions Wronskian determinant yields