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Three Theorems on the Growth of Entire Transcendental Solutions of Algebraic Differential Equations

Published online by Cambridge University Press:  20 November 2018

Sh. Strelitz
Sh. Strelitz*
Affiliation:
University of Haifa, Haifa, Israel
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G. Polya [4] has posed the problem as to whether there are entire transcendental functions of order zero satisfying an algebraic differential equation with rational coefficients. G. Polya himself showed that this is impossible for a first order algebraic differential equation. The general problem is now completely solved. G. Valiron demonstrated an example of a third order algebraic differential equation with an entire transcendental solution of order zero (Theorem 1); V. V. Zimogljad (Theorem 2) proved that every entire transcendental solution of a second order algebraic differential equation is of a positive order. It seems to us expedient to bring these results all together. We give here a proof of Theorem 2 different from and in our view simpler than that of V. V. Zimogljad. Theorem 3 refines the results of G. Polya (and of others, see for example [10]) and establishes an exact lower bound for the order of an arbitrary entire transcendental solution satisfying a first order algebraic differential equation.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 6 , 01 December 1983 , pp. 1110 - 1128
Copyright
Copyright © Canadian Mathematical Society 1983

References

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