Brisebarre, Nicolas; Habsieger, Laurent Sur les fonctions entières à double pas récurrent. (Entire functions with double recurrent step.). (French) Zbl 0927.39004 Ann. Inst. Fourier 49, No. 2, 653-671 (1999). J.-P. Bézivin and F. Gramain [Ann. Inst. Fourier 46, No. 2, 465-491 (1996; Zbl 0853.39001)] have proved that, if \(f\) is an entire function solution of the system of difference equations \[ \sum_{m=0}^M P_m(z) f(z+m\alpha)= 0,\qquad \sum_{n=0}^N Q_n(z) f(z+n\beta)=0, \tag{\(*\)} \] where \((P_m)\) and \((Q_n)\) are finite sequences of \(\mathbb{C}[z]\), \(\alpha\) and \(\beta\) are complex numbers, then \(f\) is a quotient of an exponential polynomial and a polynomial. In this paper the previous result is extended to the case when coefficients in \((*)\) are constants and \(\alpha/ \beta\) belongs to \(\mathbb{R}\setminus\mathbb{Q}\) and also for the case of functions from \(C^\infty (\mathbb{R})\). In the last part of the paper an algorithm for finding the entire solutions of \((*)\) is given. Reviewer: P.Talpalaru (Iaşi) Cited in 6 Documents MSC: 39A10 Additive difference equations 39A70 Difference operators 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 39B32 Functional equations for complex functions Keywords:difference equations; exponential polynomials; linear differential equations; \(P\)-recursive functions Citations:Zbl 0853.39001 PDFBibTeX XMLCite \textit{N. Brisebarre} and \textit{L. Habsieger}, Ann. Inst. Fourier 49, No. 2, 653--671 (1999; Zbl 0927.39004) Full Text: DOI Numdam EuDML References: [1] [Abr] , Rational solutions of linear difference and q-difference equations with polynomial coefficients, Proc. ISSAC ’95, T. Levelt ed., 1995, 285-289. ACM Press, New-York. · Zbl 0914.65131 [2] [BéGra1] et , Solutions entières d’un système d’équations aux différences, Ann. Inst. Fourier, 43-3 (1993), 791-814. · Zbl 0796.39006 [3] [BéGra2] et , Solutions entières d’un système d’équations aux différences II, Ann. Inst. Fourier, 46-2 (1996), 465-491. · Zbl 0853.39001 [4] [Bri] , An algorithm for finding integer solutions of systems of difference equations, prépublication. [5] [10] , , Shifted Jack polynomials · Zbl 0108.27503 [6] [Gra1] , Sur le théorème de Fukasawa-Gel’fond, Inv. Math., 63 (1981), 495-506. · Zbl 0461.10028 [7] [Gra2] , Équations aux différences et polynômes exponentiels, C. R. Acad. Sciences Paris, 313 (1991), 131-134. · Zbl 0736.30020 [8] [GraMi] et , Fonctions entières arithmétiques, Approximations diophantiennes et nombres transcendants, Luminy 1982, D. Bertrand et M. Waldschmidt eds, Progress in Math. 31, 1983, 99-124, Birkhäuser, Boston. · Zbl 0541.10029 [9] [Loe] , Sur certaines équations aux différences associées à des groupes, prépublication. [10] [Mas] , On certain functional equations in several variables, Approximations diophantiennes et nombres transcendants, Luminy 1982, D. Bertrand et M. Waldschmidt eds, Progress in Math. 31, 1983, 173-190, Birkhäuser, Boston. · Zbl 0549.32002 [11] [PWZ] , , , A = B, A.K. Peters, Wellesley, Massachusetts, 1996. [12] [Wal] , Nombres transcendants, Springer Lecture Notes in Math. 402, Berlin, 1974. · Zbl 0302.10030 [13] [Zei] , A holonomic systems approach to special functions identities, J. Comp. Appl. Math., 32 (1990), 321-368. · Zbl 0738.33001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.