The poles and growth of solutions of systems of complex difference equations

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R E S E A R C H

Open Access

The poles and growth of solutions of systems

of complex difference equations

Hua Wang

*

_{, Hong-Yan Xu and Bing-Xiang Liu}

*_{Correspondence:}

hhhlucy2012@126.com Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China

Abstract

In view of the Nevanlinna theory, we study the growth and poles of solutions of some classes of systems of complex diﬀerence equations and obtain some interesting results such as the lower bounds for Nevanlinna lower order, a counting function of poles and maximum modulus for solutions of such systems. They extend some results concerning functional equations to the systems of functional equations in the ﬁelds of complex equations.

MSC: 39A50; 30D35

Keywords: growth; diﬀerence equation; pole

1 Introduction and main results

The purpose of this paper is to study some properties of the poles and growth of meromor-phic solutions of the systems of complex diﬀerence equations. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions are used (see [–]). In this paper, a meromorphic function means being mero-morphic in the whole complex planeC; for a meromorphic functionf,S(r,f) denotes any quantity satisfyingS(r,f) =o(T(r,f)) for allroutside a possible exceptional setEof ﬁnite logarithmic measurelimr→∞[,r)∩E

dt

t <∞, and a meromorphic functiona(z) is called a

small function with respect tof ifT(r,a(z)) =S(r,f) =o(T(r,f)).

In , Shimomura [] and Yanagihara [] studied some existence of solutions of dif-ference equations and obtained some theorems as follows.

Theorem .(see [, Theorem .]) For any non-constant polynomial P(y),the diﬀerence equation

y(z+ ) =Py(z)

has a non-trivial entire solution.

Theorem .(see [, Corollary ]) For any non-constant rational function R(y),the dif-ference equation

y(z+ ) =Ry(z)

has a non-trivial meromorphic solution in the complex plane.

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It was proposed that the existence of suﬃciently many meromorphic solutions of ﬁnite order would be a strong indicator of integrability of an equation (see [–]).

In , Ablowitz, Halburd and Herbst [] studied some classes of complex diﬀerence equations

f(z+ ) +f(z– ) = a(z) +a(z)f+· · ·+ap(z)f

p

b(z) +b(z)f+· · ·+bq(z)fq

, ()

f(z+ )f(z– ) =a(z) +a(z)f+· · ·+ap(z)f

p

b(z) +b(z)f+· · ·+bq(z)fq

, ()

where the coeﬃcients are meromorphic functions, and obtained the following results.

Theorem .(see []) If diﬀerence equation() (or())with polynomial coeﬃcients ai(z),

bi(z)admits a transcendental meromorphic solution of ﬁnite order,then d=max{p,q} ≤.

In , Heittokangaset al.[] further investigated some complex diﬀerence equations which are similar to () and () and obtained the following results which are improvements of Theorems . and ..

Theorem .(see [, Proposition  and Proposition ]) Let c, . . . ,cn∈C\{}.If the

equa-tions

n

i=

f(z+ci) =R

z,f(z),

n

i=

f(z+ci) =R

z,f(z),

Rz,f(z):= P(z,f(z)) Q(z,f(z))=

a(z) +a(z)f +· · ·+as(z)fs

b(z) +b(z)f +· · ·+bt(z)ft

with rational coeﬃcients ai(z),bi(z)admit a transcendental meromorphic solution of ﬁnite

order,then d=max{s,t} ≤n.

In the same paper, some results of the lower bound for the characteristic functions, poles and maximum modulus of transcendental meromorphic solutions of some complex dif-ference equations are obtained as follows.

Theorem .(see [, Theorem ]) Let c, . . . ,cn∈C\{}and let m≥.Suppose y is a

transcendental meromorphic solution of the diﬀerence equation

n

i=

ai(z)y(z+ci) = m

i=

bi(z)y(z)i

with rational coeﬃcients ai(z),bi(z).Denote C:=max{|c|, . . . ,|cn|}.

() Ifyis entire or has ﬁnitely many poles,then there exist constantsK> andr>  such that

logM(r,y)≥Kmr/C

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() Ifyhas inﬁnitely many poles,then there exist constantsK> andr> such that

n(r,y)≥Kmr/c

holds for allr≥r.

Theorem .(see [, Theorem ]) Let c, . . . ,cn∈C\{}and suppose that y is a

non-rational meromorphic solution of

n

i=

di(z)y(z+ci) =

a(z) +a(z)f+· · ·+ap(z)fp

b(z) +b(z)f+· · ·+bq(z)fq

, ()

where all coeﬃcients in()are of growth o(T(r,y))without an exceptional set as r→ ∞, and di’s are non-vanishing.If d=max{p,q}>n,then for anyε( <ε< (d–n)/(d+n)),there

exists an r> such that

T(r,y)≥K

d n

 –ε

 +ε

r/C

for all r≥r,where C:=max{|c|, . . . ,|cn|}and K> is a constant.

Recently, a number of papers have focused on difference equations, difference product andq-difference in the complex planeC, and considerable attention has been paid to the growth of solutions of difference equations, value distribution and uniqueness of differ-ences analogues of Nevanlinna’s theory [, –].

In , Gao [–] also investigated the growth and existence of meromorphic so-lutions of some systems of complex diﬀerence equations and obtained some existence theorems and estimates on the proximity function and the counting function of solutions of some systems of complex diﬀerence equations.

Inspired by the ideas of Refs. [–] and Ref. [], we investigate the growth and poles of meromorphic solutions of some systems of complex diﬀerence equations and obtain the following results.

Theorem . Suppose that(f,f)is a transcendental meromorphic solution of a system of

diﬀerence equations of the form

⎧ ⎨ ⎩

n

j=aj(z)f(z+cj) =

d

i=bi(z)f(z)i,

n

j=aj(z)f(z+cj) =

d

i=bi(z)f(z)i,

()

where dd ≥ and the coeﬃcients atj(z), bti(z) (t= , )are rational functions. Denote

C:=max{|c|, . . . ,|cn|}.If ft(t= , )are entire or have ﬁnitely many poles,then there exist

constants Kt>  (t= , )and r> such that for all r≥r

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Theorem . Suppose that(f,f)is a transcendental meromorphic solution of a system of

Remark . Since system () is a particular case of system (), from the conclusions of Theorem ., we can get the following result.

Under the assumptions of Theorem ., ifft (t= , ) have inﬁnitely many poles, then

there exist constantsKt>  (t= , ) andr>  such that for allr≥r,

n(r,ft)≥Kt(dd)r/(C), t= , .

Theorem . Suppose that(f,f)is a transcendental meromorphic solution of a system of

complex diﬀerence equations of the form

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2 Some lemmas

Lemma .(Valiron-Mohon’ko []) Let f(z)be a meromorphic function.Then for all ir-reducible rational functions in f,

Rz,f(z)=

m

i=ai(z)f(z)i

n

j=bj(z)f(z)j

,

with meromorphic coeﬃcients ai(z),bj(z),the characteristic function of R(z,f(z))satisﬁes

Tr,Rz,f(z)=dT(r,f) +O(r),

where d=max{m,n}and(r) =maxi,j{T(r,ai),T(r,bj)}.

Lemma .(see []) Let f,f, . . . ,fnbe meromorphic functions.Then

T

r, λ∈I

fiλ

 f iλ_

 · · ·f iλn n

≤σ

n

i=

T(r,fi) +logs,

where I={iλ,iλ, . . . ,iλn}is an index set consisting of s elements,andσ=maxλ∈I{iλ+iλ+

· · ·+iλn}.

Lemma .(see []) Let g: (, +∞)→R,h: (, +∞)→Rbe monotone increasing func-tions such that g(r)≤h(r)outside of an exceptional set of ﬁnite logarithmic measure.Then for anyα> ,there exists r> such that g(r)≤h(αr)for all r>r.

Lemma .(see [, Lemma ]) Given> and a meromorphic function y,the Nevanlinna characteristic function T satisﬁes

Tr,y(z±)≤( +ε)Tr+ ,y(z)+κ

for all r≥/,for some constantκ.

3 The proof of Theorem 1.7

Since the coeﬃcientsat_j(z),bt_i(z) (t= , ) are rational functions, we can rewrite () as

⎧ ⎨ ⎩

n

j=Aj(z)f(z+cj) =

d

i=Bi(z)f(z)i,

n

j=Aj(z)f(z+cj) =

d

i=Bi(z)f(z)i,

()

where the coeﬃcientsAt

j(z),Bti(z) (t= , ) are polynomials.

Next, two cases will be considered as follows.

Case . Since (f,f) is a transcendental solution of system () or () and ft (t= , )

are entire, set pt

i =degAtj (j= , , . . . ,n),qti=degBti (i= , , . . . ,di),t= , , takingmt=

max{pt

, . . . ,ptn}+ , we have that

⎧ ⎨ ⎩

M(r,d

i=Bi(z)f(z)i) =M(r,

n

j=Aj(z)f(z+cj))≤nrmM(r+C,f),

M(r,d

i=Bi(z)f(z)i) =M(r,

n

j=Aj(z)f(z+cj))≤nrmM(r+C,f),

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when r is suﬃciently large. Since ft are transcendental entire functions and Bti (i=

, , . . . ,dt; t= , ) are polynomials, we have M(r,

d–

i= Bif(z)i) =o(M(r,f(z)d)) and

M(r,d–

i= Bif(z)i) =o(M(r,f(z)d)). Thus, for suﬃciently larger, we have

⎧ ⎨ ⎩

M(r,d

i=Bi(z)f(z)i)≥_M(r,Bdf(z)

d_),

M(r,d

i=Bi(z)f(z)i)≥_M(r,B_d_f(z)d).

()

From () and (), we have

⎧ ⎨ ⎩

logM(r+C,f)≥dlogM(r,f) +g(r),

logM(r+C,f)≥dlogM(r,f) +g(r),

()

where|gt(r)|<Ktlogr,t= ,  for some constantsKt>  and suﬃciently larger. From (),

for suﬃciently larger, we have

logM(r+ C,f)≥ddlogM(r,f) +g(r+C) +dg(r). ()

Iterating (), we have

logM(r+ kC,f)≥(dd)klogM(r,f) +Ek(r) +Ek(r) (k∈N), ()

where

E_k(r)=(dd)k–g(r+C) + (dd)k–g(r+ C) +· · ·+g

r+ (k– )C

≤K(dd)k– k

j=

log[r+ (j– )C] (dd)j

≤K(dd)k–

∞

j=

log[r+ (j– )C] (dd)j–

,

and

E_k(r)=d(dd)k–g(r) +d(dd)k–g(r+ C) +· · ·+dg

r+ (k– )C

≤Kd(dd)k– k

j=

log[r+ (j– )C] (dd)j–

≤Kd(dd)k–

∞

j=

log[r+ (j– )C] (dd)j–

.

Sincelog[r+kC]≤logr×logkCfor suﬃciently largerandk, and sincedd≥, we know

that the series∞_j_=log[r+(j–)C]

(dd)j– and

∞ j=

log[r+(j–)C]

(dd)j– are convergent. Thus, for suﬃciently larger, we have

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whereK_t>  (t= , ) are some constants. Sincefis a transcendental entire function, for

suﬃciently larger, we have

logM(r,f)≥K logr, ()

whereK >max{K_,K_}. Hence, from ()-(), there existsr≥esuch that forr≥r, we

have

logM(r+ kC,f)≥K(dd)klogr. ()

Choosingr∈[r,r+C) and lettingk→ ∞for each choice ofr, and for each suﬃciently

largeR:=r+ kC≥R:=r+C, we have

R∈r+ kC,r+ (k+ )C

, i.e. k>R–r–C

C . ()

From () and (), we have

logM(R,f)≥logM(r+ kC,f)≥K(dd)klogr≥K (dd)R/(C),

whereK =K(dd)

–r_–C

C _log_r__.

By using the same argument as above, we can get that there exist constantsK>  and r>  such that for allR≥r,

logM(R,f)≥K(dd)R/(C). ()

Case . Suppose that (f,f) is a solution of system () andft (t= , ) are

meromor-phic functions with ﬁnitely many poles. Then there exist polynomials Pt(z) such that

gt(z) =Pt(z)ft(z) (t= , ) are entire functions. Substitutingft(z) = g_Pt_t(₍z_z)₎ into () and again

multiplying away the denominators, we can get a system similar to (). By using the same argument as above, we can obtain that for suﬃciently larger≥r≥r,

logM(r,ft) =logM(r,gt) +logM

r,  Pt(z)

≥K_t –ε(dd)r/(C)≥Kt (dd)r/(C),

whereK_t (> ) (t= , ) are some constants.

From Case  and Case , this completes the proof of Theorem ..

Suppose that (f,f) is a solution of system () andft(t= , ) are transcendental. Since the

coeﬃcients ofPt(z,ft(z)),Qt(z,ft(z)) are rational functions, we can choose a suﬃciently

large constantR(> ) such that the coeﬃcients ofPt(z,ft(z)),Qt(z,ft(z)) have no zeros or

poles in{z∈C:|z|>R}. Sinceft(t= , ) have inﬁnitely many poles, we can choose a pole

zoff of multiplicityτ ≥ satisfying|z|>R. Then the right-hand side of the second

equation in system () has a pole of multiplicitydτ atz. Then there exists at least one

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byz+cjin the ﬁrst equation of (), we have

n

j=

a_j(z+cj)f(z+cj+cj) =R

z+cj,f(z+cj)

. ()

We now have two possibilities as follows.

(i) Ifz+cjis a pole or a zero of the coeﬃcients ofR(z,f(z)), then this process will be terminated and we can choose another polezoffin the way we did above.

(ii) Ifz+cjis neither a pole nor a zero of the coeﬃcients ofR(z,f(z)), thus the right-hand side of () has a pole of multiplicitydτatz+cj, then there exists at least one indexj_∈ {, , . . . ,n}such thatz+cj+cj_is a pole offof multiplicityτ≥dτ≥ddτ.

Replacingzbyz+cj+cj_in the second equation of (), we have

n

j=

a_j(z+cj+cj_)f(z+cj+cj_+cj) =R

z+cj+cj_,f(z+cj+cj_)

.

We proceed to follow the step above. Since the coeﬃcients ofRt(z,ft(z)) have ﬁnitely

many zeros and poles in{z∈C:|z|>R}andf has inﬁnitely many poles again, we may

construct polesζk:=z+cj+cj_+cj+cj_+· · ·+cjk+cjk(ji∈ {, , . . . ,n},ji∈ {, , . . . ,n},

i= , , . . . ,k) offof multiplicityτksatisfyingτk≥(dd)kτask→ ∞,k∈N. Since|ζk| → ∞ask→ ∞, for suﬃciently largek, sayk≥kand anyR∈[|z|,|z|+C), we have

τ(dd)k≤τ

 +dd+· · ·+ (dd)k

≤n|ζk|,f

=n|z|+ kC,f

≤nR+ kC,f

. ()

If we can choose a polezoffof multiplicityτ ≥ satisfying|z|>R, similar to the above

discussion, we can get that for suﬃciently largekand anyR∈[|z|,|z|+C),

τ (dd)k≤n

|z|+ kC,f

≤n(R+ kC,f). ()

Thus, for each suﬃciently largeR:=R+ kC≥r:=|z|+ (k+ )C, there exists a

k∈Nsuch thatR∈[|z|+ kC,|z|+ (k+ )C) (orR∈[|z|+ kC,|z|+ (k+ )C)), by

using the same method as in the proof of Theorem ., from () (or ()), we have

n(R,f)≥τ(dd)k≥τ(dd) R–|z|–C

C ≥_K_(_d__d_)R/(C), ()

or

n(R,f)≥τ(dd)k≥τ(dd) R–|z|–C

C ≥_K

(dd)R/(C), ()

whereK=τ(dd)

–|z_|–C

C _and_K_₌_τ ₍_d__d_₎

–|z_|–C

C _.

Thus, from () and (), this completes the proof of Theorem ..

From the assumptions of Theorem . and ft (t = , ) are transcendental, applying

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√

Thus, this completes the proof of Theorem ..

6 The Proof of Theorem 1.10

From the assumptions of Theorem ., by using the same argument as in Theorem ., for any givenε( <ε<

outside of a possible exceptional set of ﬁnite logarithmic measure. It follows that

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outside of a possible exceptional set of ﬁnite logarithmic measure. From () and Lemma ., it follows that for everyα> , there existsr>  such that

Tαr+ C,ft

≥ dd( –ε)

nnσσ( +ε)

T(r,ft) =:ζT(r,ft), t= , , ()

andT(r,ft) >  holds for allr≥r, whereζ > . Inductively, for any positive integerk∈N

andr≥r, from (), we have

T

αkr+α

k_{– }

α_{– }C,ft

≥ζkT(r,ft), t= , . ()

By using the same argument as in [, Theorem .], we can get thatμ(ft) =∞(t= , )

easily.

Thus, this completes the proof of Theorem ..

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

HW and HYX completed the main part of this article, HYX, HW and BXL corrected the main theorems. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors thank the referee for his/her valuable suggestions to improve the present article. This work was supported by the NSFC (61202313), the Natural Science Foundation of Jiang-Xi Province in China (Grant No. 2010GQS0119, No. 20122BAB201016, No. 20122BAB201044) and the National Science and Technology Support Plan (2012BAH25F02).

Received: 7 January 2013 Accepted: 5 March 2013 Published: 26 March 2013 References

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doi:10.1186/1687-1847-2013-75