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Advances in Difference Equations

, 2012:203 | Cite as

Results on meromorphic solutions of linear difference equations

  • Sheng Li
  • Baoqin ChenEmail author
Open Access
Research
Part of the following topical collections:
  1. Progress in Functional Differential and Difference Equations

Abstract

In this paper, we investigate meromorphic solutions of linear difference equations and prove a number of results. We give estimates for the growth of meromorphic solutions under some special cases and provide some examples to show that the answer to a question of Laine and Yang is not always positive. The zeros, poles and fixed points of finite order solutions are also studied.

MSC:39A13, 39A22, 30D35.

Keywords

difference equations growth fixed points 

1 Introduction and results

In this paper, a meromorphic function means meromorphic in the complex plane, and we assume the reader is familiar with the basic notions of Nevanlinna theory (see, e.g., [1, 2, 3]). We use the notations ρ ( f ) Open image in new window, λ ( f ) Open image in new window, λ ( 1 / f ) Open image in new window to denote the order of growth of f, the exponent of convergence of the poles of f and the exponent of convergence of the zeros of f, respectively, and we define them as follows:

Thirty years ago, some results on the existence of meromorphic solutions for certain difference equations were proved by Shimomura [4] and Yanagihara [5].

Recently, numbers of papers (see, e.g., [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]) are devoted to considering the complex difference equations and difference analogues of Nevanlinna theory. For the growth of meromorphic solutions of difference equations, Chiang and Feng [8, 9] considered the polynomial coefficients case and got

Theorem A [8, 9]

Let A 0 ( z ) , A 1 ( z ) , , A n ( z ) Open image in new window be polynomials such that there exists an integer l, 0 l n Open image in new window, such that
deg ( A l ) > max 0 j n , j l { deg ( A j ) } . Open image in new window
Suppose that f ( z ) Open image in new window is a meromorphic solution to
A n ( z ) f ( z + n ) + + A 1 ( z ) f ( z + 1 ) + A 0 ( z ) f ( z ) = 0 , Open image in new window
(1.1)

then we have ρ ( f ) 1 Open image in new window.

The following result shows that the polynomial coefficients in Theorem A can be extended to rational functions.

Theorem 1.1 Let A 0 ( z ) , A 1 ( z ) , , A n ( z ) Open image in new window be rational functions having no common zeros or poles. For j = 0 , , n Open image in new window, set A j ( z ) = p j ( z ) / q j ( z ) Open image in new window, where p j ( z ) Open image in new window, q j ( z ) Open image in new window are irreducible polynomials. If there exists an integer l, 0 l n Open image in new window, such that
deg ( p l ) deg ( q l ) > max 0 j n , j l { deg ( p j ) deg ( q j ) } , Open image in new window

then, for any meromorphic solution f ( z ) Open image in new window to (1.1), we have ρ ( f ) 1 Open image in new window.

Remark Set B j = p j k = 0 n q k / q j Open image in new window, then we see that B 0 ( z ) , B 1 ( z ) , , B n ( z ) Open image in new window are all polynomials and
deg ( B l ) > max 0 j n , j l { deg ( B j ) } . Open image in new window

And hence Theorem 1.1 follows from Theorem A. We omit the details of its proof.

For the case that some coefficients are transcendental meromorphic functions, the following two results were proved by Chaing and Feng [8] and Laine and Yang [16], respectively.

Theorem B [8]

Let A 0 ( z ) , A 1 ( z ) , , A n ( z ) Open image in new window be entire functions such that there exists an integer l, 0 l n Open image in new window, such that
ρ ( A l ) > max 0 j n , j l ρ ( A j ) . Open image in new window

If f ( z ) Open image in new window is a meromorphic solution to (1.1), then ρ ( f ) ρ ( A l ) + 1 Open image in new window.

Theorem C [16]

Let A 0 ( z ) , A 1 ( z ) , , A n ( z ) Open image in new window be entire functions of finite order such that among those coefficients having the maximal order ρ : = max 0 j n ρ ( A j ) Open image in new window, exactly one has its type strictly greater than the others. Then, for any meromorphic solution f ( z ) Open image in new window to (1.1), we have ρ ( f ) ρ + 1 Open image in new window.

In Theorems B and C, there is always some dominating coefficient A l Open image in new window such that ρ ( A l ) > 0 Open image in new window. A natural question is what happens if the dominating coefficient A l Open image in new window is of order zero? Another question raised by Laine and Yang in [16] is whether all meromorphic solutions f ( z ) Open image in new window of (1.1) satisfy ρ ( f ) max 0 j n ρ ( A j ) + 1 Open image in new window, even if there is no dominating coefficient. For the first question, we get the following result.

Theorem 1.2 Let A 0 ( z ) , A 1 ( z ) , , A n ( z ) Open image in new window be meromorphic functions such that there exists an integer l, 0 l n Open image in new window, such that A l ( z ) Open image in new window is a transcendental entire function, while A j ( z ) Open image in new window, j l Open image in new window, are all rational functions. If f ( z ) Open image in new window is a meromorphic solution to (1.1), then ρ ( f ) ρ ( A l ) + 1 Open image in new window.

Considering Laine and Yang’s question, we get the following example which indicates that the answer to their question is not always positive.

Example For a given positive integer k, f ( z ) = e z Open image in new window is an entire solution of the equation
( e z k 2 + 1 ) f ( z + 2 ) e z k 1 f ( z + 1 ) e 2 f ( z ) = 0 . Open image in new window

In this example, the relationship between ρ ( f ) Open image in new window and max 0 j n ρ ( A j ) + 1 = k + 1 Open image in new window exactly depends on k.

However, the answer may be positive in some special case. In fact, we prove the following results, in which there is still some coefficient dominating in some angle.

Theorem 1.3 Let k be a positive integer, p be a nonzero real number and f ( z ) Open image in new window be a nonconstant meromorphic solution of the difference equation
A n f ( z + n ) + + A 2 f ( z + 2 ) + ( A 1 e p z k + B 1 ) f ( z + 1 ) + ( A 0 e p z k + B 0 ) f ( z ) = 0 , Open image in new window
(1.2)

where A 0 , A 1 , , A n , B 0 , B 1 Open image in new window are all entire functions such that A 0 A 1 0 Open image in new window and max { ρ ( B 0 ) , ρ ( B 1 ) , ρ ( A j ) : 0 j n } = σ < k Open image in new window. Then we have ρ ( f ) k + 1 Open image in new window.

Theorem 1.4 Under exactly one of assumptions for the coefficients of (1.1) in Theorems  A-C and Theorems 1.1 and 1.2, if f ( z ) Open image in new window is a finite order meromorphic solution to (1.1), then λ ( f z ) = ρ ( f ) Open image in new window. What is more, either ρ + 1 ρ ( f ) max { λ ( f ) , λ ( 1 / f ) } + 1 Open image in new window or ρ ( f ) = ρ + 1 > max { λ ( f ) , λ ( 1 / f ) } + 1 Open image in new window, where ρ : = max 0 j n ρ ( A j ) Open image in new window.

Theorem 1.5 Under the assumption for the coefficients of (1.2) in Theorem  1.3, if f ( z ) Open image in new window is a finite order meromorphic solution to (1.2), then λ ( f z ) = ρ ( f ) Open image in new window. What is more, either k + 1 ρ ( f ) max { λ ( f ) , λ ( 1 / f ) } + 1 Open image in new window or ρ ( f ) = k + 1 > max { λ ( f ) , λ ( 1 / f ) } + 1 Open image in new window.

The following examples show the sharpness of the estimates in Theorems 1.4 and 1.5.

Examples (1) The gamma function Γ ( z ) Open image in new window is a meromorphic solution to the equation
f ( z + 1 ) z f ( z ) = 0 , Open image in new window

which satisfies the assumptions in Theorem A and Theorem 1.1. We see that λ ( Γ z ) = ρ ( Γ ) Open image in new window and 1 = ρ ( Γ ) < λ ( 1 / Γ ) + 1 Open image in new window .

(2) f 1 ( z ) = z e z 2 Open image in new window and f 2 ( z ) = z e z 2 sin ( 2 π z ) Open image in new window are entire solutions to the equation
z e 2 z f ( z + 1 ) e ( z + 1 ) f ( z ) = 0 , Open image in new window

which satisfies the assumptions in Theorems B and C and Theorem 1.2. We have λ ( f 1 z ) = ρ ( f 1 ) Open image in new window, λ ( f 2 z ) = ρ ( f 2 ) Open image in new window and ρ ( f 1 ) = 2 > λ ( f 1 ) + 1 Open image in new window, ρ ( f 2 ) = 2 = λ ( f 2 ) + 1 Open image in new window.

(3) f 1 ( z ) = e z 2 sin 2 π z Open image in new window and f 2 ( z ) = e z 2 Open image in new window are entire solutions of the equation
e z f ( z + 1 ) e e z f ( z ) = 0 , Open image in new window

which satisfies the assumptions in Theorem 1.3. We have λ ( f 1 z ) = ρ ( f 1 ) Open image in new window, λ ( f 2 z ) = ρ ( f 2 ) Open image in new window and ρ ( f 1 ) = 2 = 1 + λ ( f 1 ) Open image in new window and ρ ( f 2 ) = 2 > λ ( f 2 ) + 1 Open image in new window.

2 Proof of Theorem 1.2

We first recall a key lemma used to prove Theorems A and B and the pointwise estimates for difference quotient which are counterparts to Gundersen’s logarithmic derivative estimates [17] (see [8], Corollary 2.6, Theorem 8.3).

Lemma 2.1 [8]

Let f ( z ) Open image in new window be a meromorphic function of finite order ρ, ε be a positive constant, η 1 Open image in new window and η 2 Open image in new window be two distinct nonzero complex constants. Then
m ( r , f ( z + η 1 ) f ( z + η 2 ) ) = O ( r ρ 1 + ε ) , Open image in new window
and there exists a subset E ( 1 , + ) Open image in new window of finite logarithmic measure such that, for all z satisfying | z | = r [ 0 , 1 ] E Open image in new window, and as r Open image in new window sufficiently large,
exp { r ρ 1 + ε } | f ( z + η 1 ) f ( z + η 2 ) | exp { r ρ 1 + ε } . Open image in new window

Proof of Theorem 1.2 If ρ ( A l ) > 0 Open image in new window, the assertion follows from Theorem B. We next consider the case that ρ ( A l ) = 0 Open image in new window.

Assume that ρ ( f ) < 1 Open image in new window, then by Lemma 2.1 (or [7], Lemma 3.3), it is easy to deduce that for j l Open image in new window, there exists a subset E ( 1 , + ) Open image in new window of finite logarithmic measure such that, for all z = r e i θ Open image in new window, θ [ 0 , 2 π ) Open image in new window, | z | = r [ 0 , 1 ] E Open image in new window, and as r Open image in new window,
f ( z + j ) f ( z + l ) 1 . Open image in new window
(2.1)
For j l Open image in new window, set A j ( z ) = p j ( z ) / q j ( z ) Open image in new window, where p j ( z ) Open image in new window, q j ( z ) Open image in new window are irreducible polynomials. Denote d = max { 0 , deg ( p j ) deg ( q j ) : 0 j n , j l } Open image in new window. Since A l Open image in new window is a transcendental entire function, for sufficiently large r, we have
M ( r , A l ) > r d + 2 . Open image in new window
(2.2)
Now we choose a sequence z k = r k e i θ k Open image in new window, θ k [ 0 , 2 π ) Open image in new window, | z k | = r k [ 0 , 1 ] E Open image in new window such that | A l ( z k ) | = M ( r k , A l ) Open image in new window, r k Open image in new window as k Open image in new window. Combining (1.1), (2.1) and (2.2), we get
r k d + 2 < M ( r k , A l ) = | A l ( z k ) | j l | A j ( z k ) f ( z k + j ) f ( z k + l ) | < n r k d + 1 , Open image in new window

a contradiction. Our proof is thus finished. □

3 Proof of Theorem 1.3

Our tools to prove Theorem 1.3 include the following Lemma 3.1 in which the upper bound is given by Gundersen [17] while the lower bound by Chen [18].

Lemma 3.1 [17, 18]

Let f ( z ) Open image in new window be a meromorphic function with finite order ρ. Then, for any given ε > 0 Open image in new window, there exists a set E ( 1 , + ) Open image in new window of finite linear measure such that, for all z satisfying | z | = r [ 0 , 1 ] E Open image in new window and r sufficiently large,
exp { r ρ + ε } | f ( z ) | exp { r ρ + ε } . Open image in new window
Proof of Theorem 1.3 Without loss of generality, we assume that p = 1 Open image in new window. Suppose that (1.2) admits a nontrivial entire solution f ( z ) Open image in new window such that ρ ( f ) = ρ < k + 1 Open image in new window. Then by Lemma 2.1, for any given ε such that 0 < 2 ε < max { k + 1 ρ , k σ } Open image in new window, we have
exp { r ρ 1 + ε } | f ( z + j ) f ( z ) | exp { r ρ 1 + ε } , j = 1 , , n , Open image in new window
(3.1)

for all r outside of a possible exceptional set E 1 Open image in new window with finite logarithmic measure.

Applying Lemma 3.1, we have

for all r outside of a possible exceptional set E 2 Open image in new window with finite linear measure.

Choose an infinite positive real sequence z t = r t E 1 E 2 Open image in new window such that r t Open image in new window as t Open image in new window, and we get from (1.2), (3.1)-(3.3) that
exp { r t k } = | e z t k | = | j = 2 n A j ( z t ) A 0 ( z t ) f ( z t + j ) f ( z t ) + A 1 ( z t ) e z t k + B 1 ( z t ) A 0 ( z t ) f ( z t + 1 ) f ( z t ) + B 0 ( z t ) A 0 ( z t ) | < ( n + 2 ) exp { 2 r t σ + ε + r t ρ 1 + ε } , Open image in new window

a contradiction. And hence we have ρ ( f ) k + 1 Open image in new window. □

4 Proofs of Theorems 1.4 and 1.5

Lemma 4.1 [3]

Let f j ( z ) Open image in new window ( j = 1 , 2 , , n Open image in new window, n 2 Open image in new window) be meromorphic functions and g j ( z ) Open image in new window ( j = 1 , 2 , , n Open image in new window, n 2 Open image in new window) be entire functions such that
  1. 1.

    j = 1 n f j ( z ) e g j ( z ) 0 Open image in new window,

     
  2. 2.

    g j ( z ) g k ( z ) Open image in new window are not constant functions for 1 j < k n Open image in new window.

     
  3. 3.

    T ( r , f j ) = o ( T ( r , e g h g k ) ) Open image in new window ( r Open image in new window, r E Open image in new window), where E is an exceptional set of finite linear measure, 1 j n Open image in new window and 1 h < k n Open image in new window.

     

Then f j ( z ) 0 Open image in new window ( j = 1 , 2 , , n Open image in new window).

Proofs of Theorems 1.4 and 1.5 In fact, we only give the proof of Theorem 1.5 since the proof of Theorem 1.4 is similar.

Suppose that f ( z ) Open image in new window is a nonconstant meromorphic solution of (1.2) such that ρ ( f ) < Open image in new window. We firstly prove that λ ( f z ) = ρ ( f ) Open image in new window. Submitting f ( z ) = g ( z ) + z Open image in new window into (1.2), we get
A n g ( z + n ) + + A 2 g ( z + 2 ) + ( A 1 e p z k + B 1 ) g ( z + 1 ) + ( A 0 e p z k + B 0 ) g ( z ) = D ( z ) , Open image in new window
where
D ( z ) = z { A n ( z ) + + A 2 ( z ) + ( A 1 ( z ) e p z k + B 1 ( z ) ) + ( A 0 ( z ) e p z k + B 0 ( z ) ) } 0 . Open image in new window

Obviously, we have ρ ( D ) k Open image in new window.

Now, for any given ε > 0 Open image in new window, applying Lemma 2.1, we can deduce that
m ( r , 1 g ) = m ( r , A n g ( z + n ) + + ( A 1 e p z k + B 1 ) g ( z + 1 ) D ( z ) g ( z ) + A 0 e p z k + B 0 ) j = 1 n m ( r , g ( z + j ) g ( z ) ) + j = 0 n T ( r , A j ) + T ( r , B 0 ) + T ( r , B 1 ) + T ( r , e p z k ) + T ( r , e p z k ) + T ( r , D ) + O ( log r ) = O ( r ρ 1 + ε ) = S ( r , g ) . Open image in new window
This implies that
N ( r , 1 f z ) = N ( r , 1 g ) = T ( r , g ) + S ( r , g ) = T ( r , f ) + S ( r , f ) . Open image in new window

Then λ ( f z ) = ρ ( f ) Open image in new window follows.

Next, we assert that either k + 1 ρ ( f ) max { λ ( f ) , λ ( 1 / f ) } + 1 Open image in new window or ρ ( f ) = k + 1 Open image in new window. If the assertion does not hold, we have max { k , λ ( f ) , λ ( 1 / f ) } + 1 < ρ ( f ) < Open image in new window.

Assume that z = 0 Open image in new window is a zero (or a pole) of f ( z ) Open image in new window of order m. Applying the Hadamard factorization of a meromorphic function, we write f ( z ) Open image in new window as follows:
f ( z ) = z m P 1 ( z ) P 2 ( z ) e Q ( z ) , Open image in new window

where P 1 ( z ) Open image in new window, P 2 ( z ) Open image in new window are entire functions such that ρ ( P 1 ) = λ ( f ) Open image in new window, ρ ( P 1 ) = λ ( 1 / f ) Open image in new window and Q ( z ) Open image in new window is a polynomial such that deg Q ( z ) = q > max { k , λ ( f ) , λ ( 1 / f ) } + 1 Open image in new window.

Now, we obtain from (1.2) that
j = 1 h j ( z ) e Q ( z + j 1 ) = 0 , Open image in new window
(4.1)
where
h 1 = ( A 0 e z k + B 0 ) z m P 1 ( z ) P 2 ( z ) , h 2 = ( A 1 e z k + B 1 ) ( z + 1 ) m P 1 ( z + 1 ) P 2 ( z + 1 ) Open image in new window
and
h j = ( z + j 1 ) m P 1 ( z + j 1 ) P 2 ( z + j 1 ) Open image in new window

for 3 j n + 1 Open image in new window. Notice that deg ( Q ( z + h ) Q ( z + k ) ) = q 1 > ρ ( h j ) Open image in new window for 1 h < k n Open image in new window and 0 j n Open image in new window. Thus, Lemma 4.1 is valid for (4.1) and hence we get that h j ( z ) 0 Open image in new window for j = 0 , 1 , , n Open image in new window, a contradiction to our assumption. This completes our proof. □

Notes

Acknowledgements

The authors would like to thank the editor and the referees for their constructive comments to improve the readability of our paper. This work is supported by the National Natural Science Foundation of China (No. 11226091) and the Natural Science Research Projects of GDOU (No. 1212331).

References

  1. 1.
    Hayman WK: Meromorphic Functions. Clarendon, Oxford; 1964.zbMATHGoogle Scholar
  2. 2.
    Laine I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin; 1993.CrossRefzbMATHGoogle Scholar
  3. 3.
    Yang CC, Yi HX Math. Appl. 557. In The Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht; 2003.CrossRefGoogle Scholar
  4. 4.
    Shimomura S: Entire solutions of a polynomial difference equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 1981, 28: 253–266.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Yanagihara N: Meromorphic solutions of some difference equations. Funkc. Ekvacioj 1980, 23: 309–326.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ablowitz M, Halburd RG, Herbst B: On the extension of the Painlevé property to difference equations. Nonlinearity 2000, 13: 889–905. 10.1088/0951-7715/13/3/321MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bergweiler W, Langley JK: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc. 2007, 142: 133–147. 10.1017/S0305004106009777MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chiang YM, Feng SJ:On the Nevanlinna characteristic f ( z + η ) Open image in new window and difference equations in the complex plane. Ramanujan J. 2008, 16: 105–129. 10.1007/s11139-007-9101-1MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chiang YM, Feng SJ: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans. Am. Math. Soc. 2009, 361(7):3767–3791. 10.1090/S0002-9947-09-04663-7MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Halburd RG, Korhonen RJ: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 2006, 314: 477–487. 10.1016/j.jmaa.2005.04.010MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Halburd RG, Korhonen RJ: Existence of finite-order meromorphic solutions as a detector of integrability in difference equations. Physica D 2006, 218: 191–203. 10.1016/j.physd.2006.05.005MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 2006, 31: 463–478.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Halburd RG, Korhonen RJ: Finite-order meromorphic solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc. 2007, 94: 443–474.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Halburd RG, Korhonen RJ: Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations. J. Phys. A, Math. Theor. 2007, 40: R1-R38. 10.1088/1751-8113/40/6/R01MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Heittokangas J, Korhonen R, Laine I, Rieppo J, Tohge K: Complex difference equations of Malmquist type. Comput. Methods Funct. Theory 2001, 1: 27–39.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Laine I, Yang CC: Clunie theorems for difference and q -difference polynomials. J. Lond. Math. Soc. 2007, 76(2):556–566.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gundersen GG: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. Lond. Math. Soc. 1988, 37(2):88–104.MathSciNetzbMATHGoogle Scholar
  18. 18.
    Chen ZX: The growth of solutions of a class of second-order differential equations with entire coefficients. Chin. Ann. Math., Ser. B 1999, 20(1):7–14. in Chinese 10.1142/S0252959999000035CrossRefMathSciNetzbMATHGoogle Scholar

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© Li and Chen; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Authors and Affiliations

  1. 1.College of ScienceGuangdong Ocean UniversityZhanjiangChina

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