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Abstract. When/(z) is given by a known power series expansion, it is possible to construct the power series expansion for/(z; p) = e~p:f(z). We define popl to be the value of p for which the expansion for/(z; p) converges most rapidly. When/(z) is an entire function of order 1, we show that pofl is uniquely defined and may be characterized in terms of the set of singularities z, = I/o, of an associated function h(z). Specifically, it is the center of the smallest circle in the complex plane which contains all points a¡.

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Summary. The purpose of this paper is to construct a generalization of the Euler-Knopp transformation. Using this, one may recover previously known transformations, derive new transformations useful for numerical calculations and derive generating functions and other formulas of theoretical interest involving well-known functions.

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