TY - JOUR

T1 - On solutions of differential equations which satisfy certain algebraic relations

AU - Sperber, Steven

PY - 1986/9

Y1 - 1986/9

N2 - In the following, we provide another proof (Theorem 3.1 below) of recent results of Harris-Sibuya, using some elementary commutative algebra. Our purpose is to give a uniform treatment for their results which also permits some generalization. We note that the study of differential equations under the hypothesis that the solutions satisfy an algebraic relation is not new. Fano, among others, made a systematic study of this situation in the last century. Also Lamé equations in which two solutions have a rational function as their product have proved to be a good source of examples for unusual arithmetic behavior. But in the case of Harris-Sibuya, as well as the present paper, the solutions need not be solutions of the same linear equation. In the treatment below the differential equation only enters in dilineating a type of recursion.

AB - In the following, we provide another proof (Theorem 3.1 below) of recent results of Harris-Sibuya, using some elementary commutative algebra. Our purpose is to give a uniform treatment for their results which also permits some generalization. We note that the study of differential equations under the hypothesis that the solutions satisfy an algebraic relation is not new. Fano, among others, made a systematic study of this situation in the last century. Also Lamé equations in which two solutions have a rational function as their product have proved to be a good source of examples for unusual arithmetic behavior. But in the case of Harris-Sibuya, as well as the present paper, the solutions need not be solutions of the same linear equation. In the treatment below the differential equation only enters in dilineating a type of recursion.

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U2 - 10.2140/pjm.1986.124.249

DO - 10.2140/pjm.1986.124.249

M3 - Article

AN - SCOPUS:4344685400

VL - 124

SP - 249

EP - 256

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -