FINITE LOGARITHMIC ORDER SOLUTIONS OF LINEAR q-DIFFERENCE EQUATIONS

Title & Authors
FINITE LOGARITHMIC ORDER SOLUTIONS OF LINEAR q-DIFFERENCE EQUATIONS
Wen, Zhi-Tao;

Abstract
During the last decade, several papers have focused on linear q-difference equations of the form $\small{{\sum}^n_{j=0}a_j(z)f(q^jz)=a_{n+1}(z)}$ with entire or meromorphic coefficients. A tool for studying these equations is a q-difference analogue of the lemma on the logarithmic derivative, valid for meromorphic functions of finite logarithmic order $\small{{\rho}_{log}}$. It is shown, under certain assumptions, that $\small{{\rho}_{log}(f)}$ = max$\small{{{\rho}_{log}(a_j)}}$ + 1. Moreover, it is illustrated that a q-Casorati determinant plays a similar role in the theory of linear q-difference equations as a Wronskian determinant in the theory of linear differential equations. As a consequence of the main results, it follows that the q-gamma function and the q-exponential functions all have logarithmic order two.
Keywords
logarithmic Borel exceptional value;logarithmic derivative;logarithmic exponent of convergence;logarithmic order;q-Casorati determinant;q-difference equation;
Language
English
Cited by
1.
Growth of Meromorphic Solutions to Some Complex Functional Equations, Computational Methods and Function Theory, 2016, 16, 3, 489
2.
Growth of Meromorphic Solutions of Finite Logarithmic Order of Linear Difference Equations, Fasciculi Mathematici, 2015, 54, 1
References
1.
M. H. Abu Risha, M. H. Annaby, M. E. H. Ismail, and Z. S. Mansour, Linear q-difference equations, Z. Anal. Anwend. 26 (2007), no. 4, 481-494.

2.
G. E. Andrews, R. Asker, and R. Roy, Special Functions, Cambridge University press, 1999.

3.
D. C. Barnett, R. G. Halburd, W. Morgan, and R. J. Korhonen, Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), no. 3., 457-474.

4.
C. Berg and H. L. Pedersen, Logarithmic order and type of indeterminate moment problems, With an appendix by Walter Hayman, Difference equations, special functions and orthogonal polynomials, 51-79, World Sci. Publ., Hackensack, NJ, 2007.

5.
W. Bergweiler and W. K. Hayman, Zeros of solutions of a functional equation, Comput. Methods Funct. Theory 3 (2003), no. 1, 55-78.

6.
W. Bergweiler, K. Ishizaki, and N. Yanagihara, Meromorphic solutions of some functional equations, Methods Appl. Anal. 5 (1998), no. 3, 248-258.

7.
W. Bergweiler, K. Ishizaki, and N. Yanagihara, Growth of meromorphic solutions of some functional equations, I. Aequationes Math. 63 (2002), no. 1-2, 140-151.

8.
B.-Q. Chen and Z.-X. Chen, Meromorphic solutions of some q-difference equations, Bull. Korean Math. Soc. 48 (2011), no. 6, 1303-1314.

9.
B.-Q. Chen, Z.-X. Chen, and S. Li, Properties on solutions of some q-difference equations, Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 10, 1877-1886.

10.
Y.-M. Chiang and S. J. Feng, On the Nevanlinna characteristic of f(z+${\eta}$) and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105-129.

11.
Peter T.-Y. Chern, On meromorphic functions with finite logarithmic order, Trans. Amer. Math. Soc. 358 (2006), no. 2, 473-489.

12.
J. Clunie, On integral functions having prescribed asymptotic growth, Canad. J. Math. 17 (1965), 396-404.

13.
J. Clunie and T. Kovari, On integral functions having prescribed asymptotic growth. II, Canad. J. Math. 20 (1968), 7-20.

14.
G. Gasper and M. Rahman, Basic Hypergeometric Series, Second Edition, Cambridge, Cambridge Univ. Press 2004.

15.
A. A. Goldberg and I. V. Ostrovskii, Value distribution of meromorphic functions, Translated from the 1970 Russian original by Mikhail Ostrovskii.With an appendix by Alexandre Eremenko and James K. Langley.Translations of Mathematical Monographs, 236. American Mathematical Society, Providence, RI, 2008.

16.
R. Halburd, R. Korhonen, and K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, arXiv:0903.3236v1.

17.
W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.

18.
W. K. Hayman, Slowly growing integral and subharmonic functions, Comment. Math. Helv. 34 (1960), 75-84.

19.
J. Heittokangas, I. Laine, J. Rieppo, and D.-G. Yang, Meromorphic solutions of some linear functional equations, Aequations Math. 60 (2000), no. 1-2, 148-166.

20.
I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin-New York, 1993.

21.
H. Meschkowski, Differenzengleichungen, Studia Mathematica, Bd. XIV Vandenhoeck and Ruprecht, Gottingen 1959.