R E S E A R C H
Open Access
On the meromorphic solutions of some linear
difference equations
Huifang Liu
1*and Zhiqiang Mao
2**Correspondence: [email protected]; [email protected] 1Institute of Mathematics and Informatics, Jiangxi Normal University, Nanchang, 330022, China 2School of Mathematics and Computer, Jiangxi Science and Technology Normal University, Nanchang, 330038, China
Abstract
This paper is devoted to studying the growth of meromorphic solutions of some linear difference equations. We obtain some results on the growth of meromorphic solutions when most coefficients in such equations have the same order, which are supplements of previous results due to Chiang and Feng, and Laine and Yang. Some examples are given to show the sharpness of our results.
MSC: 39B32; 30D35; 39A10
Keywords: complex difference equation; meromorphic solution; growth
1 Introduction and main results
In this paper, the term meromorphic function will mean meromorphic in the whole com-plex planeC. It is assumed that the reader is familiar with the standard notations and basic results of Nevanlinna theory (see,e.g., [–]). In addition, we useσ(f) andσ(f) to
denote the order and the hyper-order of a meromorphic functionf(z), andλ(f) andλ(/f) to denote the exponents of convergence of zeros and poles off(z), respectively. For a mero-morphic functionf(z), when <σ(f) <∞or <σ(f) <∞, its typeτ(f) and hyper-type τ(f) are defined byτ(f) =limr→∞Tr(σr(,ff)) andτ(f) =limr→∞
logT(r,f)
rσ(f) (see,e.g., [, , ]).
Recently, meromorphic solutions of complex difference equations have become a sub-ject of great interest from the viewpoint of Nevanlinna theory, due to the apparent role of the existence of such solutions of finite order for the integrability of discrete difference equations (see,e.g., [–]). Halburd and Korhonen [] proved that when the difference equation
ω(z+ ) +ω(z– ) =R(z,ω), (.)
whereR(z,ω) is rational in both of its arguments and has an admissible meromorphic solution of finite order, then eitherωsatisfies a difference Riccati equation, or equation (.) can be transformed by a linear change inω(z) to a difference Painlevé equation or a linear difference equation. Thus the linear difference equation plays an important role in the study of properties of difference equations.
Chiang and Feng [] considered the linear difference equation
ak(z)f(z+k) +· · ·+a(z)f(z+ ) +a(z)f(z) = , (.)
and obtained the following results.
Theorem A[] Let a(z), . . . ,ak(z)be polynomials.If there exists an integer l(≤l≤k) such that
deg(al) >max
≤j≤k j=l
deg(aj)
holds,then every meromorphic solution f (≡)of Eq. (.)satisfiesσ(f)≥,wheredeg(al) denotes the degree of the polynomial al.
Theorem B[] Let a(z), . . . ,ak(z)be entire functions.If there exists an integer l(≤l≤n) such that
σ(al) >max
≤j≤k j=l
σ(aj)
holds,then every meromorphic solution f (≡)of Eq. (.)satisfiesσ(f)≥σ(al) + .
Note that in Theorems A and B, Eq. (.) has only one dominating coefficiental. For the
case when there is no dominating coefficient and all coefficients are polynomials in Eq. (.), Chen [] obtained an improvement of Theorem A.
Theorem C[] Let a(z), . . . ,ak(z)be polynomials such that
deg(a+· · ·+ak) =max
≤j≤k{degaj} ≥.
Then every finite order meromorphic solution f (≡)of Eq. (.)satisfiesσ(f)≥. For the case when there is more than one of coefficients which have the maximal order, Laine and Yang [] obtained the following result.
Theorem D[] Let a(z), . . . ,ak(z)be entire functions of finite order such that among those having the maximal orderσ=max≤j≤k{σ(aj)},exactly one has its type strictly greater than the others.Then for every meromorphic solution f (≡)of Eq. (.),we haveσ(f)≥σ+ .
Note that in Theorem D, the condition that exactly one coefficient has the maximal type among those coefficients having the maximal order, guarantees that every meromorphic solutionf (≡) of Eq. (.) satisfiesσ(f)≥max≤j≤k{σ(aj)}+ . The following example
shows that when there exists more than one coefficient having the maximal type among those coefficients having the maximal order,σ(f)≥max≤j≤k{σ(aj)}+ may hold.
Example . The difference equation
ez+f(z+ ) –e–zf(z) =
admits an entire solutionf(z) =e–z, wherea
(z) =ez+,a(z) = –e–zsatisfyσ(a) =σ(a) =
=σ,τ(a) =τ(a). Hereσ(f) = =σ+ .
Question . What can be said if there exists more than one coefficient having the max-imal type and the maxmax-imal order in Eq. (.)?
Question . What can be said if all coefficients of Eq. (.) have the order zero? From
the definition of the type of an entire function and the assumptions of Theorem B or The-orem D, we know that in TheThe-orems B and D there exists at least one coefficiental such
thatσ(al) > .
Question . What can be said if there exists more than one coefficient having the order
∞in Eq. (.)?
The main purpose of this paper is to investigate the above questions for Eq. (.). The remainder of the paper investigates the properties of meromorphic solutions of a non-homogeneous linear difference equation corresponding to (.).
Theorem . Let aj(z) =Aj(z)ePj(z)(j= , , . . . ,k),where Pj(z) =αjnzn+· · ·+αjare polyno-mials with degree n(≥),Aj(z) (≡)are entire functions ofσ(Aj) <n.Ifαjn(j= , , . . . ,k) are distinct complex numbers,then every meromorphic solution f(≡)of Eq. (.)satisfies σ(f)≥max≤j≤k{σ(aj)}+ .
Theorem . Let aj(z) =Aj(z)ePj(z)+Dj(z) (j= , , . . . ,k),where Aj(z),Pj(z)satisfy the
hy-pothesis of Theorem.,Dj(z)are entire functions withσ(Dj) <n.Ifαjn(j= , , . . . ,k)are distinct complex numbers,then every meromorphic solution f (≡)of Eq. (.)satisfies σ(f)≥max≤j≤k{σ(aj)}+ .
Remark . In Theorems . and ., we haveσ(a) =· · ·=σ(ak) andτ(a) =· · ·=τ(ak)
if|αn|=· · ·=|αkn|. Therefore Theorems . and . are supplements of Theorem D.
Remark . From the proof of Theorems . and ., we know that the same result also holds for Eq. (.) in the case when at least two coefficients have the form ofaj(z) in
The-orem . or ., and the orders of the others are less thann.
Theorem . Let H be a complex set satisfying dens{r =|z| : z ∈ H} > , and let a(z), . . . ,ak(z)be entire functions satisfyingmax≤j≤k{σ(aj)} ≤σ.If there exists an inte-ger l(≤l≤k)such that for some constants≤α<βandδ> sufficiently small,
al(z)≥exp
βrσ–δ, (.)
aj(z)≤exp
αrσ–δ, j= , . . . ,k,j=l, (.)
as z→ ∞for z∈H,then every meromorphic solution f (≡)of Eq. (.)satisfiesσ(f)≥
σ(al) + .
Remark . Note thatσmay be zero in Theorem ..
Example . The difference equation
ze–z–+e–f(z+ ) –ez–+zf(z) =
admits a solution f(z) =e(z+)
, where a(z) =ze–z– +e–, a(z) = –(ez–+z)
sat-isfy the hypothesis of Theorem . and σ(a) =σ(a), τ(a) =τ(a). Here σ(f) = =
max{σ(a),σ(a)}+ .
When there exists more than one coefficient having the order∞in Eq. (.), we obtain the following result. Note that in this case Theorem D is invalid.
Theorem . Let a,a, . . . ,ak be entire functions.If there exists an integer l(≤l≤k) such that
maxσ(aj) :j= , . . . ,k,j=l
≤σ(al)
<σ(al) <∞
,
maxτ(aj) :σ(aj) =σ(al)
<τ(al)
<τ(al) <∞
,
then every meromorphic solution f (≡)of Eq. (.)satisfiesσ(f) =∞andσ(f)≥σ(al).
Next we consider the properties of meromorphic solutions of the non-homogeneous linear difference equation corresponding to (.)
ak(z)f(z+k) +· · ·+a(z)f(z+ ) +a(z)f(z) =F(z), (.)
whereF(z) (≡) is an entire function.
Theorem . Let aj(z) (j= , . . . ,k)satisfy the hypothesis of Theorem.or Theorem.,
and let F(z)be an entire function ofσ(F) <n.Then at most one meromorphic solution f of Eq. (.)satisfiesmax≤j≤k{σ(aj)} ≤σ(f) <max≤j≤k{σ(aj)}+ andmax{λ(f),λ(/f)}= σ(f),the other solutions f satisfyσ(f)≥max≤j≤k{σ(aj)}+ .
Theorem . Let aj(z) (j= , . . . ,k)satisfy the hypothesis of Theorem.,and let F(z)be an entire function.Then
(i) Ifσ(F) <σ(al)orσ(F) =σ(al),τ(F) <τ(al),then every meromorphic solutionf
(≡)of Eq.(.)satisfiesσ(f) =∞andσ(f)≥σ(al).
(ii) Ifσ(F) >σ(al),then every meromorphic solutionf (≡)of Eq.(.)satisfies σ(f) =∞andσ(f)≥σ(F).
2 Lemmas
Lemma .[] Letη,ηbe two arbitrary complex numbers,and let f(z)be a meromorphic function of finite orderσ.Letε> be given,then there exists a subset E⊂(, +∞)with finite logarithmic measure such that for all|z|=r∈/E∪[, ],we have
exp–rσ–+ε≤f(z+η) f(z+η)
Lemma .[] Suppose that P(z) = (α+iβ)zn+· · · (α,βare real numbers,|α|+|β| = ) is a polynomial with degree n≥,A(z) (≡)is an entire function withσ(A) <n.Set g(z) =
A(z)eP(z),z=reiθ,δ(P,θ) =αcosnθ –βsinnθ.Then for any givenε> ,there exists a set
E⊂[, π)that has linear measure zero,such that for anyθ∈[, π)\(E∪E),there is R> such that for|z|=r>R,we have
(i) ifδ(P,θ) > ,then
exp( –ε)δ(P,θ)rn<greiθ<exp( +ε)δ(P,θ)rn;
(ii) ifδ(P,θ) < ,then
exp( +ε)δ(P,θ)rn<greiθ<exp( –ε)δ(P,θ)rn,
where E={θ∈[, π) :δ(P,θ) = }is a finite set.
Lemma .[] Let f(z)be an entire function of orderσ(f) =σ< +∞.Then for any given ε> ,there is a set E⊂[, +∞)having finite linear measure such that for all z satisfying
|z|=r∈/[, ]∪E and r sufficiently large,we have
exp–rσ+ε≤f(z)≤exprσ+ε.
Lemma .[] Let f be a non-constant meromorphic function,c∈C,δ< andε> .
Then
m
r,f(z+c)
f(z)
=o
T(r+|c|,f)+ε
rδ
(.)
for all r outside of a possible exceptional set E with finite logarithmic measure Edrr <∞. Remark . By [], we know that
+o()Tr–|h|,f(z)≤Tr,f(z+h)
≤ +o()Tr+|h|,f(z) (r>r> ), (.)
whereh∈C. So, by (.) and (.), we immediately have
m
r,f(z+c)
f(z+h)
=o
T(r+|c–h|+|h|,f)+ε
rδ
(.)
for allroutside of a possible exceptional setEwith finite logarithmic measure Edr r <∞.
Lemma . [] Let f be a meromorphic function with hyper-order <σ(f) <∞and hyper-type <τ(f) <∞,then for any givenβ <τ(f),there exists a subset E⊂[, +∞) of infinite logarithmic measure such that T(r,f) >βrσ(f)holds for all r∈E.
Proof By the Weierstrass factorization, we obtain
We discuss the following two cases.
By (.), we getσ(H)≥n. On the other hand, by the elementary order considerations, we haveσ(H)≤n. So,σ(H) =n. Then byG(z) =H(z)/Q(z) andσ(Q) <n, we getσ(G) =n.
3 Proofs of the results
Proof of Theorem. Letf (≡) be a meromorphic solution of (.). Suppose thatσ(f) <
n+ , then by Lemma ., for any givenε> , there exists a setE⊂(, +∞) with finite logarithmic measure such that for all|z|=r∈/E∪[, ], we have
ff((zz++jl))≤exprσ(f)–+ε (j= , . . . ,k,j=l). (.)
Set z=reiθ, α
jn =|αjn|eiϕj and δ(Pj,θ) =|αjn|cos(ϕj+nθ) (j= , . . . ,k). ThenE ={θ ∈
[, π) :δ(Pj,θ) = ,j= , . . . ,k} ∪ {θ∈[, π) :δ(Pj–Pi,θ) = , ≤i<j≤k}is a set of
lin-ear measure zero. Considering eachaj(z) =Aj(z)ePj(z), by Lemma ., for the aboveε> ,
there exists a setFj⊂[, π) of linear measure zero such that for anyz=reiθ satisfying θ∈[, π)\(E∪Fj) andrsufficiently large, we have
(i) ifδ(Pj,θ) > , then
exp( –ε)δ(Pj,θ)rn
<aj
reiθ<exp( +ε)δ(Pj,θ)rn
; (.)
(ii) ifδ(Pj,θ) < , then
exp( +ε)δ(Pj,θ)rn
<aj
reiθ<exp( –ε)δ(Pj,θ)rn
. (.)
Set E=
k
j=Fj, thenE is a set of linear measure zero. Sinceαjn are distinct complex
numbers, there exists only onel∈ {, . . . ,k}such thatδ(Pl,θ) =max{δ(Pj,θ) :j= , . . . ,k}
for anyθ∈[, π)\(E∪E). Now we take a rayargz=θ∈[, π)\(E∪E) such that δ(Pl,θ) > . Letδ=δ(Pl,θ),δ=max{δ(Pj,θ) :j= , . . . ,k,j=l}, thenδ>δ. We discuss
the following two cases.
Case .δ> . We rewrite (.) in the form
–al(z) = k
j=
j=l aj(z)
f(z+j)
f(z+l). (.)
By (.), (.) and (.), we get forz=reiθand sufficiently larger∈/E∪[, ],
exp( –ε)δrn
≤al
reiθ≤ k
j=
j=l
exp( +ε)δ(Pj,θ)rn
exprσ(f)–+ε
≤kexp( +ε)δrn
exprσ(f)–+ε. (.)
When < ε<min{δ–δ
δ+δ,n+ –σ(f)}, by (.), we get
exp
δ–δ
r
n
≤kexprσ(f)–+ε.
Case . δ < . By (.), (.) and (.), we get forz=reiθ and sufficiently larger∈/ E∪[, ],
exp( –ε)δrn
≤al
reiθ≤ k
j=
j=l
exp( –ε)δ(Pj,θ)rn
exprσ(f)–+ε
≤kexprσ(f)–+ε.
This is a contradiction. Hence we getσ(f)≥n+ =max≤j≤k{σ(aj)}+ . Proof of Theorem. By Lemmas . and ., we know that for any givenε> , there is a setE⊂[, +∞) having finite linear measure such that for allzsatisfying|z|=r∈/[, ]∪E
andrsufficiently large, we have
exp( –ε)δ(Pj,θ)rn
<aj
reiθ<exp( +ε)δ(Pj,θ)rn
(.)
ifδ(Pj,θ) > , and
aj
reiθ<exprσ(Dj)+ε (.)
ifδ(Pj,θ) < . Then using the similar argument to that of Theorem . and only replacing
(.) (or (.)) by (.) (or (.)), we can prove Theorem ..
Proof of Theorem. Letf (≡) be a meromorphic solution of (.). Suppose thatσ(f) <
σ + , then by Lemma ., for any givenε( <ε<σ+ –σ(f) – δ), there exists a set
E⊂(, +∞) with finite logarithmic measure such that for all|z|=r∈/E∪[, ], we have
ff((zz++lj))≤exprσ(f)–+ε<exprσ–δ (j= , . . . ,k,j=l). (.)
Rewrite (.) in the form
– =
k
j=
j=l
aj(z)f(z+j) al(z)f(z+l)
. (.)
SinceE∪[, ] has finite logarithmic measure, the density ofE∪[, ] is zero. Hence (.) and (.) also hold forz∈H\E∪[, ]. Substituting (.), (.) and (.) into (.), we get forz∈H\E∪[, ],
≤
k
j=
j=l
exp(α–β)rσ–δexprσ–δ
→,
Proof of Theorem. Letf (≡) be a meromorphic solution of (.). By (.) we get
Then by Lemma ., we know that there exists a setH of infinite logarithmic measure such that
holds for sufficiently largern.
Hence by (.), for sufficiently largern, we have Proof of Theorem . Let f (≡ ) be a meromorphic solution of (.). Suppose that
σ(f) <max≤j≤k{σ(aj)}, then by Lemma . we obtainσ(F) =σ(
phic solution of the homogeneous linear difference equation corresponding to (.), and
σ(f–f) <max≤j≤k{σ(aj)}+ . By Theorem . or Theorem ., we get a contradiction.
. Substituting (.) into (.), we get
k
by using a similar to the above argument, we also obtain a contradiction.
≤Tr,F(z)+ +o()Tr+|l|,f(z)+
other hand, for sufficiently largernwe have
T(rn,F) <exp
By the definition of hyper-order, we know that there exists a sequence{rn}such thatrn→
∞, and for any givenε( < ε<σ(F) –σ(al)) and sufficiently largern, we have
The authors declare that they have no competing interests.
Authors’ contributions
Both authors drafted the manuscript, read and approved the final manuscript.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11201195, 11171119), the Natural Science Foundation of Jiangxi, China (No. 20122BAB201012, 20132BAB201008, 20122BAB211005), the STP of the Education Department of Jiangxi, China (No. GJJ12179). The authors thank the referee for his/her valuable suggestions to improve the present article.
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doi:10.1186/1687-1847-2013-133