Solutions entières d’un système d’équations aux différences. II. (Entire functions solutions of a system of difference equations. II.). (French) Zbl 0853.39001
Summary: [For part I see this journal 43, No. 3, 791-814 (1993; Zbl 0796.39006).]
Let \(s\) be a rational positive integer, and \(f\) an entire function in \(s\) complex variables. In a previous article we have shown that if \(f\) satisfies a system of linear difference equations with polynomial coefficients in two different directions, assuming \(s=1\) and a condition on the equations, then \(f\) is the quotient of an exponential polynomial by a polynomial. In this paper, we remove the restrictive condition and prove the same result for \(s \geq 2\).
Let \(s\) be a rational positive integer, and \(f\) an entire function in \(s\) complex variables. In a previous article we have shown that if \(f\) satisfies a system of linear difference equations with polynomial coefficients in two different directions, assuming \(s=1\) and a condition on the equations, then \(f\) is the quotient of an exponential polynomial by a polynomial. In this paper, we remove the restrictive condition and prove the same result for \(s \geq 2\).
MSC:
| 39A10 | Additive difference equations |
| 39B32 | Functional equations for complex functions |
| 34M99 | Ordinary differential equations in the complex domain |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
Citations:
Zbl 0796.39006References:
| [1] | [BéGra] et , Solutions entières d’un système d’équations aux différences, Ann. Inst. Fourier, Grenoble, 43-3 (1993), 791-814. · Zbl 0796.39006 |
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