Asymptotic analysis for

linear difference equations

Author:
Katsunori Iwasaki

Journal:
Trans. Amer. Math. Soc. **349** (1997), 4107-4142

MSC (1991):
Primary 39A10, 39A12, 40A05, 46M20

DOI:
https://doi.org/10.1090/S0002-9947-97-01849-7

MathSciNet review:
1401775

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Abstract | References | Similar Articles | Additional Information

Abstract: We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly. Finally we obtain Gevrey estimates for the solutions. In a forthcoming paper we will develop the theory of cohomology groups for recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.

**1.**Benzaid, Z. and Lutz, D.A.,*Asymptotic representation of solutions of perturbed systems of linear difference equations*, Studies in Appl. Math.**77**(1987), 195-221. MR**90f:39003****2.**Birkhoff, G.D.,*General theory of linear difference equations*, Trans. Amer. Math. Soc.**12**(1911), 243-284.**3.**Charrière, H. and Gérard, R.,*The rings of formal and convergent inverse factorial series*, Kumamoto J. Math.**5**(1992), 1-20. MR**94b:39019****4.**Coffman, C.V.,*Asymptotic behavior of solutions of ordinary difference equations*, Trans. Amer. Math. Soc.**110**(1964), 22-51. MR**27:6054****5.**Evgrafov, M.A.,*On the asymptotic behavior of solutions of difference equations*(in Russian), Dokl. Akad. Nauk SSSR**121**(1958), 26-219. MR**21:5102****6.**Gérard, R. and Lutz, D.A.,*Convergent factorial series solutions of singular operator equations*, Analysis**10**(1990), 99-145. MR**92f:39005****7.**Hirai, I. and Yanagihara, N.,*Difference equations in Banach spaces*, Comm. Math. Univ. St. Paul**28**(1980), 51-61. MR**81k:39002****8.**Ishizuka, S.,*On solution complexes of -modules associated with confluent hypergeometric differential equations*, (in Japanese), Master thesis, Ochanomizu University, Tokyo, 1994.**9.**Ishizuka, S., Iwasaki, K. and Majima, H.,*Gevrey cohomology groups for the Humbert systems*, in preparation.**10.**Iwasaki, K.,*Cohomology groups for recurrence relations*, Preprint (1996).**11.**Li, Z.-H.,*The asymptotic estimates of solutions of difference equations*, J. Math. Anal. Appl.**94**(1983), 181-192. MR**94g:39001****12.**Milne-Thomson, L.M.,*The calculus of finite differences*, Chelsea Publ., New York, 1981. MR**13:245c (2nd ed.)****13.**Nielsen, N.,*Recherches sur les séries de factorielles*, Ann. Ecole Norm. Sup.**19**(1902), 409-453.

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Additional Information

**Katsunori Iwasaki**

Affiliation:
Department of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153 Japan

Address at time of publication:
Department of Mathematics, Kyushu University, G-10-1 Hakozaki, Higashi-ku, Fukuoka 812-81 Japan

Email:
iwasaki@ms.u-tokyo-ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-97-01849-7

Keywords:
Difference equation,
profile operator,
factorial asymptotic expansion,
Gevrey estimate

Received by editor(s):
October 31, 1994

Received by editor(s) in revised form:
March 18, 1996

Article copyright:
© Copyright 1997
American Mathematical Society