Order reduction method for linear difference equations
Authors:
R. Korhonen and O. Ronkainen
Journal:
Proc. Amer. Math. Soc. 139 (2011), 32193229
MSC (2010):
Primary 39A06; Secondary 30D35, 39A10, 39A12
Published electronically:
April 13, 2011
MathSciNet review:
2811278
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Additional Information
Abstract: An order reduction method for homogeneous linear difference equations, analogous to the standard order reduction of linear differential equations, is introduced, and this method is applied to study the Nevanlinna growth relations between meromorphic coefficients and solutions of linear difference equations.
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N. M. Ruijsenaars, First order analytic difference equations and
integrable quantum systems, J. Math. Phys. 38 (1997),
no. 2, 1069–1146. MR 1434226
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 1.
 M. J. Ablowitz, R. G. Halburd, and B. Herbst, On the extension of the Painlevé property to difference equations, Nonlinearity 13 (2000), 889905. MR 1759006 (2001g:39003)
 2.
 Z.X. Chen and K. W. Shon, The growth of solutions of differential equations with coefficients of small growth in the disc, J. Math. Anal. Appl. 297 (2004), no. 1, 285304. MR 2080381 (2005g:34224)
 3.
 Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic of and difference equations in the complex plane, Ramanujan J. 16 (2008), 105129. MR 2407244 (2009c:30073)
 4.
 , On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions, Trans. Amer. Math. Soc. 361 (2009), no. 7, 37673791. MR 2491899 (2010c:30044)
 5.
 Y. M. Chiang and S. N. M. Ruijsenaars, On the Nevanlinna order of meromorphic solutions to linear analytic difference equations, Stud. Appl. Math. 116 (2006), no. 3, 257287. MR 2220338 (2006m:39005)
 6.
 A. A. Goldberg and I. V. Ostrovskii, Value distribution of meromorphic functions, Translations of Mathematical Monographs, vol. 236, American Mathematical Society, Providence, RI, 2008. MR 2435270 (2009f:30067)
 7.
 G. G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), 415429. MR 920167 (88j:34013)
 8.
 G. G. Gundersen, E. M. Steinbart, and S. Wang, The possible orders of solutions of linear differential equations with polynomial coefficients, Trans. Amer. Math. Soc. 350 (1998), 12251247. MR 1451603 (98h:34006)
 9.
 R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), 477487. MR 2185244 (2007e:39030)
 10.
 R. G. Halburd, R. J. Korhonen, and K. Tohge, Holomorphic curves with shiftinvariant hyperplane preimages, preprint (arXiv:0903.3236), 2009.
 11.
 W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford, 1964. MR 0164038 (29:1337)
 12.
 J. Heittokangas, R. Korhonen, and J. Rättyä, Linear differential equations with coefficients in weighted Bergman and Hardy spaces, Trans. Amer. Math. Soc. 360 (2008), no. 2, 10351055. MR 2346482 (2008k:34338)
 13.
 K. Ishizaki and N. Yanagihara, WimanValiron method for difference equations, Nagoya Math. J. 175 (2004), 75102. MR 2085312 (2006c:39001)
 14.
 L. Kinnunen, Linear differential equations with solutions of finite iterated order, Southeast Asian Bull. Math. 22 (1998), no. 4, 385405. MR 1811183 (2001i:34154)
 15.
 I. Laine, Nevanlinna theory and complex differential equations, Walter de Gruyter, Berlin/New York, 1993. MR 1207139 (94d:34008)
 16.
 C. Praagman, Fundamental solutions for meromorphic linear difference equations in the complex plane, and related problems, J. Reine Angew. Math. 369 (1986), 101109. MR 850630 (88b:39004)
 17.
 S. N. M. Ruijsenaars, First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), no. 2, 10691146. MR 1434226 (98m:58065)
 18.
 J. M. Whittaker, Interpolatory function theory, Cambridge Tracts in Mathematics and Mathematical Physics, vol. 33, StechertHafner, New York, 1964. MR 0185330 (32:2798)
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Additional Information
R. Korhonen
Affiliation:
Department of Physics and Mathematics, University of Eastern Finland, Joensuu Campus, P. O. Box 111, FI80101 Joensuu, Finland
Email:
risto.korhonen@uef.fi
DOI:
http://dx.doi.org/10.1090/S000299392011110815
PII:
S 00029939(2011)110815
Received by editor(s):
March 9, 2010
Published electronically:
April 13, 2011
Additional Notes:
The research reported in this paper was supported in part by the Academy of Finland grants No. 118314 and No. 134792 and the European Science Foundation RNP HCAA
Communicated by:
Ken Ono
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
