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On difference Riccati equations and second order linear difference equations. (English) Zbl 1215.39021
The author considers second order linear difference equations and the associated difference Riccati equation in the complex domain, namely $$\Delta^2y(z) + A(z)y(z)=0 $$ and $$ \Delta f(z) = {{f(z)^2+A(z)}\over{f(z)-1}}$$ where he defines $$\Delta\varphi(z)=\varphi(z+1)-\varphi(z)$$ with the standard iteration of the difference operator; $A(z)$ is a meromorphic function. He considers: meromorphic solutions of the difference Riccati equation and the discrete analogue of the cross ratio; linear second order difference equations in the complex domain and growth problems for the solutions.

39A45Difference equations in the complex domain
39A13Difference equations, scaling ($q$-differences)
39A06Linear equations (difference equations)
30D15Special classes of entire functions; growth estimates
Full Text: DOI
[1] Bank S.B., Gundersen G., Laine I.: Meromorphic solutions of the Riccati differential equation. Ann. Acad. Sci. Fenn. Ser. A I Math. 6(2), 369--398 (1982) · Zbl 0493.34007
[2] Bank S.B., Laine I.: On the zeros of meromorphic solutions and second-order linear differential equations. Comment. Math. Helv. 58(4), 656--677 (1983) · Zbl 0532.34008 · doi:10.1007/BF02564659
[3] Chiang Y.M., Feng S.J.: On the Nevanlinna characteristic of f(z + {$\eta$}) and difference equations in the complex plane. Ramanujan J. 16(1), 105--129 (2008) · Zbl 1152.30024 · doi:10.1007/s11139-007-9101-1
[4] Elaydi S.: An Introduction to Difference Equations, Third edition, Undergraduate Texts in Mathematics. Springer, New York (2005) · Zbl 1071.39001
[5] Halburd R.G., Korhonen R.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314(2), 477--487 (2006) · Zbl 1085.30026 · doi:10.1016/j.jmaa.2005.04.010
[6] Halburd R.G., Korhonen R.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31(2), 463--478 (2006) · Zbl 1108.30022
[7] Hille E.: Ordinary Differential Equations in the Complex Domain. Dover Publications, Inc., Mineola NY (1997) · Zbl 0901.34001
[8] Ishizaki K., Yanagihara N.: Wiman-Valiron method for difference equations. Nagoya Math. J. 175, 75--102 (2004) · Zbl 1070.39002
[9] Jank G., Volkmann L.: Einführung in die Theorie der Ganzen und Meromorphen Funktionen mit Anwendungen auf Differentialgleichungen. Birkhäuser Verlag, Basel-Boston (1985) · Zbl 0682.30001
[10] Kelley W.G., Peterson A.C.: Difference Equations, An Introduction with Applications, Second Edition. Harcourt/Academic Press, San Diego (2001)
[11] Kohno M.: Global Analysis in Linear Differential Equations, Mathematics and Its Applications, vol. 471. Kluwer Academic Publishers, Dordrecht (1999) · Zbl 0933.34002
[12] Laine I.: Nevanlinna Theory and Complex Differential Equations, de Gruyter Studies in Mathematics, vol. 15. Walter de Gruyter & Co., Berlin (1993) · Zbl 0784.30002
[13] Laine I., Yang C.C.: Clunie theorems for difference and q-difference polynomials. J. Lond. Math. Soc. (2) 76(3), 556--566 (2007) · Zbl 1132.30013 · doi:10.1112/jlms/jdm073
[14] Mohon’ko, A.Z., Mohon’ko, V.D.: Estimates of the Nevanlinna characteristics of certain classes of meromorphic function, and their applications to differential equations. Sibirsk. Mat. Zh. 15 1305--1322. (1974) (Russian)
[15] Yanagihara N.: Meromorphic solutions of some difference equations. Funkcial. Ekvac. 23, 309--326 (1980) · Zbl 0474.30024
[16] Ye Z.: The Nevanlinna functions of the Riemann zeta-function. J. Math. Anal. Appl. 233, 425--435 (1999) · Zbl 0924.11071 · doi:10.1006/jmaa.1999.6343