On meromorphic equivalence of linear difference operators
Annales de l'Institut Fourier, Volume 40 (1990) no. 3, p. 683-699

We consider linear difference equations whose coefficients are meromorphic at . We characterize the meromorphic equivalence classes of such equations by means of a system of meromorphic invariants. Using an approach inspired by the work of G. D. Birkhoff we show that this system is free.

On étudie des équations linéaires aux différences finies à coefficients méromorphes à l’infini. On caractérise les classes d’équivalence méromorphes de telles équations par un système d’invariants méromorphes. On démontre la liberté de ce systèmes en utilisant une méthode inspirée des travaux de G.D. Birkhoff.

@article{AIF_1990__40_3_683_0,
     author = {Immink, Gertrude K.},
     title = {On meromorphic equivalence of linear difference operators},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {40},
     number = {3},
     year = {1990},
     pages = {683-699},
     doi = {10.5802/aif.1228},
     zbl = {0697.39006},
     mrnumber = {92e:39018},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1990__40_3_683_0}
}
Immink, Gertrude K. On meromorphic equivalence of linear difference operators. Annales de l'Institut Fourier, Volume 40 (1990) no. 3, pp. 683-699. doi : 10.5802/aif.1228. http://www.numdam.org/item/AIF_1990__40_3_683_0/

[1] G.D. Birkhoff, The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations, Proc. Amer. Acad. Arts and Sci., 49 (1913), 521-568. | JFM 44.0391.03

[2] G.D. Birkhoff and W.J. Trjitzinsky, Analytic theory of linear difference equations, Acta Math., 60 (1933), 1-89. | JFM 59.0450.03 | Zbl 0006.16802

[3] J. Ecalle, Les fonctions résurgentes, t. III, Publ. Math. d'Orsay, Université de Paris-Sud (1985).

[4] A.S. Fokas and M.J. Ablowitz, On the initial value problem of the second Painlevé transcendent, Comm. Math. Phys., 91 (1983), 381-403. | MR 86b:34011 | Zbl 0524.35094

[5] G.K. Immink, Reduction to canonical forms and the Stokes phenomenon in the theory of linear difference equations, To appear in SIAM J. Math. Anal., 22 (1991). | MR 92c:39005 | Zbl 0733.39004

[6] G.K. Immink, On the asymptotic behaviour of the coefficients of asymptotic power series and its relevance to Stokes phenomena, To appear in SIAM J. Math. Anal., 22 (1991). | Zbl 0716.30032

[7] W.B. Jurkat, Meromorphe Differentialgleichungen, Lecture Notes in Mathematics 637, Springer Verlag, Berlin (1978). | MR 82a:34004 | Zbl 0408.34004

[8] B. Malgrange, Remarques sur les équations différentielles à points singuliers irréguliers, In : Equations différentielles et systèmes de Pfaff dans le champ complexe, Lecture Notes in Mathematics, 712 (1979), 77-86. | MR 80k:14019 | Zbl 0423.32014

[9] N.I. Muskhelishvili, Singular integral equations, Noordhoff, Groningen, 1953.

[10] C. Praagman, The formal classification of linear difference operators, Proc. Kon. Ned. Ac. Wet. Ser. A, 86 (1983), 249-261. | MR 85c:12006 | Zbl 0519.39003

[11] Y. Sibuya, Stokes phenomena, Bull. Amer. Math. Soc., 83 (1977), 1075-1077. | MR 56 #720 | Zbl 0386.34008

[12] E.C. Titchmarsh, The theory of functions (2nd ed.), Oxford University Press, Oxford, 1939. | JFM 65.0302.01

[13] N.P. Vekua, Systems of singular integral equations, Gordon and Breach, New York, 1967.

[14] J. Martinet, J.P. Ramis, Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Publ. Math. IHES, 55 (1982), 63-162. | Numdam | MR 84k:34011 | Zbl 0546.58038