XIV. On the convergence of infinite series of analytic functions

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In the first section of the following work an attempt is made to deal with the convergence of infinite series of functions defined by linear differential equations of the second order from the most general point of view. Functions of Lamé Bessel and Legendre are considered as examples. In the second section the results obtained are applied to the expansion of an arbitrary uniform analytic function of an arbitrary uniform analytic function of z in a series of hypergeometric functions, and the expansion is shown to be valid if the function is regular within a certain ellipse in the z -plane. An expansion in a series of Legendre’s associated functions is deduced by a transformation. The method has been applied by the writer to other cases, but the foregoing offer adequate illustration of the general theory.