Differential operators and differential equations of by Pieter Cornelis. Sikkema

By Pieter Cornelis. Sikkema

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Additional info for Differential operators and differential equations of infinite order with constant coefficients: Researches in connection with integral functions of finite order

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Consequently this power series defines n=0 a function that is analytic for I z z. 00 6) As is well-known the radius R of a power series E Cnzn is given by the formula n=n (A) R = lim inf - 1 . ICnln From this formula it is easy to deduce that also (B) R=_ 1.. lira sup I cn In n-00 Cf. Pringsheim [3], Bd IL 1, p. 244. 55 00 anxn defines a function that is Conversely, if the power series n-0 analytic for I z 15 r, its radius of convergence exceeds T. Then it follows from (A) that formula (54) holds.

86. There also results are stated concerning integral functions of finite order that are of the maximum type of the order 1 or of an order exceeding 1. Just before we were going to press it appeared from the Mathematical Reviews vol. 13 (1952) p. 929, that Gelfond [1] has generalized the notion asymptotic period for the latter functions. We had no access to Gelfond's paper. Here we shall not enter into Whittaker's and Gelfond's results, since we restrict ourselves to integral functions not exceeding the normal type of the order 1.

In chapter III we shall considerably improve on these results. In fact, we prove that, if the function y(x) is a transcendental integral function not exceeding the minimum type of the order 1, then the integral function h(x) is of the same kind as the function y(x). For instance, if the function y(x) is an integral function of the normal type r of the order a (then we now have 0 < a < I since the function y(x) does not exceed the minimum type of the order 1 (see definition 1)), we prove that the function h(x) is likewise an integral function of the normal type r of the order a.

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