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

Example text

N=0 In case 0

Pringsheim , 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. Hence (53) holds and this means that (37) with a = t is valid. From this we may deduce with the help of the reasoning that followed formula (40), that the expression G(z) in (35) with a = 1, defines an integral function not exceeding the normal type, less than r-1, of the order 1. This proves theorem I conclusively.

A -1 finzn is on account of 0 < a < 1 divergent for every value of n=o z 0 0. It is clear that for 0 < a < 1 there exists a non-enumerable infinite set of such "divergent" operators. To see this one need only replace the coefficient of D° in the differential operator just mentioned by an arbitrary constant. Theorem z. A necessary and sufficient condition for the differential 00 operator F(D) _ a,,D' to be applicable to all integral functions y(x) =o that are of the minimum type of the order a > 0, is, that the numbers a7, are such that (55) G(z) _ 2 and Z.