Solutions of the nth order Cauchy difference equation on groups

R E S E A R C H

Open Access

Solutions of the

nth-order Cauchy

difference equation on groups

Qiaoling Guo, Bolin Ma and Lin Li

*_{Correspondence: [email protected]}

College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, P.R. China

Abstract

Let (G,·) be a group, (H, +) be an Abelian group, andf:G→Hbe a function. In this paper, for a positive integern, we first give a representation ofnth-order Cauchy difference offvia the function as

C(n)f(x1,x2,. . .,xn,xn+1) =

1≤m≤n+1

(–1)n+1–m

1≤i1<i2<···<im≤n+1

f(xi1xi2· · ·xim),

wherex1,x2,. . .,xn+1∈G. Then, based on the representation, we get some special

solutions ofC(n)f= 0 on free groups. Moreover, sufficient and necessary conditions on symmetric groups and finite cyclic groups are also obtained.

MSC: 39B52; 39A70

Keywords: Cauchy difference; free group; symmetric group; cyclic group

1 Introduction

It is well known that the solutions to Jensen’s functional equation

f(x+y) +f(x–y) = f(x) (.)

are just all linear functionsf(x) =cx+d if we assume thatf are continuous, and they are the set of all homomorphisms (whend= ) on real lineRas an additive group. Let (G,·) be a group, and (H, +) be an Abelian group. Denote bye∈Gand ∈Hthe identity elements, respectively. In [], it was pointed out that the set of solutions is not equivalent to all homomorphisms on a general group. Therefore, finding out the solutions to equation (.) on groups becomes an interesting problem (see [, ] and references therein). Note that these solutions are related to their Cauchy differences [–]. For a functionf:G→H, its Cauchy differencesC(m)_{f are defined by}

C()f =f, (.)

C()f(x,x) =f(xx) –f(x) –f(x),

C(m+)f(x,x, . . . ,xm+)

=C(m)f(xx,x, . . . ,xm+) –C(m)f(x,x, . . . ,xm+) –C(m)f(x,x, . . . ,xm+). (.)

The first-order Cauchy differenceC()_{f will be abbreviated asCf. In general,C}(m+)_{f=  if} and only ifC(m)_{f ism-additive, andC}(m)_{f =  impliesC}(n)_{f=  forn≥m[]. Fischer and} Heuvers [–] mentioned that any generalized polynomialf withC(m)_{f =  has a unique} representation of the formf(x) =f(x) +· · ·+fm(x), where eachfj(x),j= , . . . ,m, is aj-form.

Besides the properties of solutions satisfying (.), we are also concerned with its general solutions. In [, ], by using the reduction formulas and relations given in [, ], Ng provided the general solution of the second-and third-order Cauchy difference equations on free groups. Some previous results on second order were extended to high order in []. Moreover, for the general solution of the third-order Cauchy difference equation on symmetric groups and finite cyclic groups, see reference [].

Plentiful results were also devoted to the generalized Cauchy difference functional equa-tions of the form

f(x) +f(y) –f(x+y) =gH(x,y), (.)

whereHis given, andf,gare unknown. Under regularity assumptions onHand a partic-ular solutionf,g, Ebanks [, ] investigated the general solution (f,g) of equation (.). For particular forms ofg(H(x,y)), the existence and stability of solutions to equation (.) are studied extensively in,e.g., [–].

Note that equations (.) and (.) are concerning functions of one variable. Also, for a mapF:Gn_{→H, we define themth partial Cauchy difference with respect to theith}

vari-able (≤i≤n), which is connected to the representation of the generalized polynomialF (see [–, ]).

It is natural to consider the general expression ofnth-order Cauchy differenceC(n)f and determine the solutions to equation

C(n)f= , (.)

which becomes the motivation of this paper. Remark that equation (.) for free group with just one generator has been solved in []. In our paper, the purpose is to determine all solutions to equation (.) on some given groups. For simplicity, the general solution to equation (.) will be denoted by

KerC(n)(G,H) =f:G→H|C(n)f = . (.)

Obviously,KerC(n)_{(G,H) is an Abelian group under the pointwise addition of functions,} andHom(G,H)⊆KerC(n)_(G,H).

2 Some properties on solutions

In this section, we study properties of solutions fornth-order Cauchy differences.

Proposition  For a positive integer n,the nth-order Cauchy difference C(n)_{f can be} ex-pressed in terms of f as

C(n)f(x,x, . . . ,xn,xn+) =

n+

m=

(–)n+–m ≤i<i<···<im≤n+

Proof Claim (.) is true forn= ,  from (.)-(.). Suppose that (.) holds forn– . Note that

C(n)f(x,x, . . . ,xn,xn+)

=C(n–)f(xx,x, . . . ,xn,xn+)

–C(n–)f(x,x, . . . ,xn,xn+) –C(n–)f(x,x, . . . ,xn,xn+)

by (.). Then, we have

C(n–)f(xx,x, . . . ,xn,xn+)

= (–)n–f(xx) + (–)n–

≤i≤n+ f(xi)

+ (–)n– ≤i≤n+

f(xxxi) + (–)

n–

≤i<i≤n+ f(xixi)

+· · ·+ (–) ≤i<i<···<in–≤n+

f(xxxixi· · ·xin–)

+ (–)

≤i<i<···<in–≤n+

f(xixi· · ·xin–)

+ (–)

≤i<i<···<in–≤n+

f(xxxixi· · ·xin–) + (–) _f(x

x· · ·xn+)

+f(xxx· · ·xn+),

C(n–)f(x,x, . . . ,xn,xn+)

= (–)n–_f(x

) + (–)n–

≤i≤n+ f(xi)

+ (–)n– ≤i≤n+

f(xxi) + (–)

n– ≤i<i≤n+

f(xixi)

+· · ·+ (–) ≤i<i<···<in–≤n+

f(xxixi· · ·xin–)

+ (–)

≤i<i<···<in–≤n+

f(xixi· · ·xin–)

+ (–)

≤i<i<···<in–≤n+

f(xxixi· · ·xin–) + (–) _f(x

x· · ·xn+)

+f(xx· · ·xn+),

and

C(n–)f(x,x, . . . ,xn,xn+)

= (–)n–f(x) + (–)n–

≤i≤n+ f(xi)

+ (–)n– ≤i≤n+

f(xxi) + (–)

n–

≤i<i≤n+

f(xixi)

+· · ·+ (–) ≤i<i<···<in–≤n+

+ (–) ≤i<i<···<in–≤n+

f(xixi· · ·xin–)

+ (–)

≤i<i<···<in–≤n+

f(xxixi· · ·xin–) + (–) _f(x

x· · ·xn+)

+f(xx· · ·xn+).

Summing over these three equalities, we get

C(n)f(x,x, . . . ,xn,xn+) =

n+

m=

(–)n+–m ≤i<i<···<im≤n+

f(xixi· · ·xim),

and this gives (.).

Proposition  If f ∈KerC(n)(G,H),then the following properties are valid.

(i) Fori= , , . . . ,n– andj= , , . . . ,i+ ,we have

C(i)f(x,x, . . . ,xj–,e,xj+, . . . ,xi+) = . (.)

In particular,

f(e) = . (.)

(ii) C(n–)_{f is a homomorphism with respect to each variable.}

Proof We first check (.). Forf ∈KerC(n)(G,H), we takex=ein (.). Then, it follows from (.) that

 =C(n)f(e,x, . . . ,xn+)

=C(n–)f(e,x,x, . . . ,xn+) –C(n–)f(x,x, . . . ,xn+) –C(n–)f(e,x, . . . ,xn+)

= (–)C(n–)f(e,x, . . . ,xn+) = (–)C(n–)f(e,x, . . . ,xn+) =· · ·

= (–)n–Cf(e,xn+) = (–)nf(e),

which gives (.).

Obviously, (.) is true fori= ,  by (.) and (.). Assume that (.) holds for all num-bers smaller thani≥. By induction, forj= , we have

C(i)f(e,x,x, . . . ,xi+)

=C(i–)f(x,x, . . . ,xi+) –C(i–)f(x,x, . . . ,xi+) –C(i–)f(e,x, . . . ,xi+)

= –C(i–)f(e,x, . . . ,xi+) = ;

forj= ,

C(i)f(x,e,x, . . . ,xi+)

=C(i–)f(x,x, . . . ,xi+) –C(i–)f(e,x, . . . ,xi+) –C(i–)f(x,x, . . . ,xi+)

and

C(i)f(x,x, . . . ,xj–,e,xj+, . . . ,xi+)

=C(i–)f(xx,x, . . . ,xj–,e,xj+, . . . ,xi+)

–C(i–)f(x,x, . . . ,xj–,e,xj+, . . . ,xi+)

–C(i–)f(x,x, . . . ,xj–,e,xj+, . . . ,xi+)

= 

in the casej≥. This confirms (.).

We infer from (.) and (.) thatC(n–)f is a homomorphism with respect to the first variable. Then by the symmetry among the variables the Cauchy difference ofC(n–)_{f in} its first variable is equivalent to the other variable, and therefore, (ii) is proved.

Remark  For any functionf :G→H, the following statements are equivalent:

(i) The functionf ∈KerC(n)_(G,H).

(ii) C(n–)_{f is a homomorphism with respect to thej-variable,j∈ {, , . . . ,n+ }.}

Next, we give two useful lemmas.

Lemma (Lemma . in []) The following identity is valid for any function f :G→H and∈N:

f(xx· · ·x) =

m≤

≤i<i<···<im≤

C(m–)f(xi,xi, . . . ,xim). (.)

Lemma (Proposition . in []) Let n be a positive integer.If f∈KerC(n)_{(G,H),then for} all x∈G and p∈Z,we have

fxp= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

n–

j=

p(p–)···(p–j)

(j+)! C(j)f(x, x, . . . ,x

j+

), p≤or p≥n,

p–

j=

p(p–)···(p–j) (j+)! C

(j)_{f(x,x, . . . ,x}

j+

),  <p<n. (.)

The following statement is a vision of Lemma  under the restrictionf∈KerC(n)_(G,H).

Theorem  Suppose that f ∈KerC(n)_{(G,H).Then the following identities are valid.}

(i) Ifl≥n,then,formk∈Zandxk∈G,i= , , . . . ,l,such thatxk=xk+,we have

fxm  x

m  · · ·x

ml

l

=

l

k= _n–

j=

mk(mk– )· · ·(mk–j)

(j+ )! C (j)_f(x

k,xk, . . . ,xk

j+ )

+

n–

i=

≤k<k<···<ki≤l

C(i–)fxmk

k ,x

mk

k , . . . ,x

m_ki ki

+mkmk· · ·mkn

≤k<k<···<kn≤l

(ii) Ifl<n,then,formk∈Zand allxk∈G,k= , , . . . ,l,such thatxk=xk+,we have

Therefore, by (.) in Lemma  and (ii) of Proposition  we have

fxmk

which is formula (.).

In the case l<n, (.) is obtained by (.)-(.) directly. This completes the whole

proof.

3 Solutions on free groups

W(x;a,b) =

i>j,ai=a,aj=b

ninj (.)

along with (.). Referred from [, ], these functions are well defined. Furthermore, they satisfy the following relations:

W(xy;a) =W(x;a) +W(y;a), (.)

W(x;a,b) =W(x;b,a). (.)

Proposition  For any fixed a∈A and fixed pair of distinct a, b inA, the following assertions hold:

(i) W(·;a)belongs toKerC(n)_{(A ,Z);}

(ii) W(·;a,b)belongs toKerC(n)(A ,Z);

(iii) W(·;a,b)belongs toKerC(n)(A ,Z).

Proof Statement (i) follows from the fact thatx→W(x;a) is a morphism fromA toZ by (.).

Now we consider statement (ii). Letx,x, . . . ,xn+in the free groupA be written as

x=ata

t · · ·a

tr_

r, x=a

t a

t · · ·a

tr_

r, . . . ,

xn+=a

tn+,

n+,a

tn+,

n+,· · ·a

tn+,_rn+

n+,rn+.

Then by (.) we have

C(n)W(x,x, . . . ,xn+;a,b)

=

n+

m=

(–)n+–m ≤l<l<···<lm≤n+

W(xlxl· · ·xlm;a,b)

= (–)n

n+

l=

W(xl;a,b) + (–)n–

≤l<l≤n+

W(xlxl;a,b)

+ (–)n– ≤l<l<l≤n+

W(xlxlxl;a,b) +· · ·

+ (–)

≤l<l<···<ln≤n+

W(xlxl· · ·xln;a,b)

+ (–)W(xx· · ·xn+;a,b)

= (–)n n+

k=

i<j,aki=a,akj=b tkitkj

+ (–)n–

≤l<l≤n+ ≤k≤

i<j,a_{lk i}=a,a_{lk j}=b

tlkitlkj+

al_i=a,al_j=b tlitlj

+ (–)n–

≤l<l<l≤n+ ≤k≤

i<j,a_{lk i}=a,a_{lk j}=b

+ ≤p<q≤

a_lpi=a,a_lqj=b

tlpitlqj

+· · ·

+ (–)

≤l<l···<ln≤n+ ≤k≤n

i<j,a_{lk i}=a,a_{lk j}=b

tlkitlkj

+

≤p<q≤n

a_lpi=a,a_lqj=b

tlpitlqj

+ (–) ≤k≤n+

i<j,aki=a,akj=b tkitkj+

≤p<q≤n+

api=a,aqj=b tpitqj

=

(–)n ≤k≤n+

i<j,aki=a,akj=b tkitkj

+ (–)n– ≤l<l≤n+

≤k≤

i<j,a_{lk i}=a,a_{lk j}=b

tlkitlkj

+ (–)n– ≤l<l<l≤n+

≤k≤

i<j,a_{lk i}=a,a_{lk j}=b

tlkitlkj+· · ·

+ (–)

≤l<l<···<ln≤n+

≤k≤n

i<j,a_{lk i}=a,a_{lk j}=b

tlkitlkj

+ (–) ≤k≤n+

i<j,aki=a,akj=b tkitkj

+

(–)n– ≤l<l≤n+

al_i=a,al_j=b tlitlj

+ (–)n– ≤l<l<l≤n+

≤p<q≤

a_lpi=a,a_lqj=b

tlpitlqj

+· · ·+ (–) ≤l<l<···<ln≤n+

≤p<q≤n

a_lpi=a,a_lqj=b

tlpitlqj

+ (–) ≤p<q≤n+

tpitqj

I+I,

where

I= (–)n

≤k≤n+

i<j,aki=a,akj=b tkitkj

+ (–)n– ≤l<l≤n+

≤k≤

i<j,a_{lk i}=a,a_{lk j}=b

tlkitlkj

+ (–)n– ≤l<l<l≤n+

≤k≤

i<j,a_{lk i}=a,a_{lk j}=b

tlkitlkj+· · ·

+ (–) ≤l<l<···<ln≤n+

≤k≤n

i<j,a_{lk i}=a,a_{lk j}=b

tlkitlkj

+ (–) ≤k≤n+

i<j,aki=a,akj=b

By the symmetry we can see that for any ≤k≤n+  andi<j, the coefficient of the item tkitkjin (.) is identical and equals

(–)n

n 

+ (–)n–

n 

+ (–)n–

n 

+· · ·+ (–)

n n– 

+ (–)

n n

= (– + )n= ,

which givesI= . Now compute

I= (–)n–

≤l<l≤n+

ali=a,alj=b tlitlj

+ (–)n– ≤l<l<l≤n+

≤p<q≤

a_lpi=a,a_lqj=b

tlpitlqj

+· · ·+ (–) ≤l<l<···<ln≤n+

≤p<q≤n

a_lpi=a,a_lqj=b

tlpitlqj

+ (–) ≤p<q≤n+

tpitqj. (.)

Obviously, for any ≤p<q≤n+ , the coefficient of the itemtpitqjin (.) is identical and

equals

(–)n–

n–  

+ (–)n–

n–  

+ (–)n–

n–  

+· · ·+ (–)

n–  n– 

+ (–)

n–  n– 

= (– + )n–= ,

which givesI= . This concludes assertion (ii).

Statement (iii) follows from (.) directly.

4 Solutions on symmetric groups

The symmetric group on a finite setXis a group whose elements are all bijective maps from X to itself and whose group operation is that of the map composition. If X=

{, , . . . ,m}, then it is called a symmetric group of degreem, denoted bySm. In this section,

we considerC(n)_{f =  forG=S}

m.

Proposition  If f ∈KerC(n)(Sm,H),then for any i= , , . . . ,n,we have

C(n–)f(x,x, . . . ,xi–,yy· · ·yp,xi+, . . . ,xn)

=C(n–)f(x,x, . . . ,xi–,yπ()yπ()· · ·yπ(p),xi+, . . . ,xn) (.)

Proof Note thatC(n–)_{f is a homomorphism with respect to each variable andH is an} Abelian group, which yields

C(n–)f(x,x, . . . ,xi–,yy· · ·yp,xi+, . . . ,xn)

Proof We only need to prove (.). To this end, we need the following facts.

=

n+  

+

n+ 



+

n+  

+· · ·+

n+  n+ 

f(τ)

= 

n+

j=

n+  j

f(τ) = 

( + )

n+_f(τ)

= nf(τ),

which confirms the even case. Whennis odd, by (.), (.) becomes

C(n)f(τ,τ, . . . ,τ)

= –

n+  

f(τ) –

n+ 



f(τ) –

n+  

f(τ) –· · ·–

n+  n

f(τ)

=

–

n+  

–

n+ 



–

n+  

–· · ·–

n+ 

n

f(τ)

= – 

n+

j=

n+  j

f(τ) = – ( + )

n+_f(τ)

= –n_f(τ).

This completes the proof.

Proposition  For any-cycleσ,σ, . . . ,σnand f ∈KerC(n)(Sm,H),we have

C(n–)f(σ,σ, . . . ,σn) =C(n–)f

(), (), . . . , (). (.)

Proof For any -cycle σ, there exists z∈Sm such that σ =z()z–. Hence, for any x,x, . . . ,xn∈Sm, by (.) we have

C(n–)f(σ,x,x, . . . ,xn)

=C(n–)fz()z–,x,x, . . . ,xn

=C(n–)f()zz–,x,x, . . . ,xn

=C(n–)f(),x,x, . . . ,xn

. (.)

Similarly, for any ≤i≤n,

C(n–)_f(x

,x, . . . ,xi–,σi,xi+, . . . ,xn)

=C(n–)fx,x, . . . ,xi–,z()z–,xi+, . . . ,xn

,

=C(n–)fx,x, . . . ,xi–, ()zz–,xi+, . . . ,xn

,

=C(n–)fx,x, . . . ,xi–, (),xi+, . . . ,xn

. (.)

In particular, (.) follows from (.)-(.).

Lemma  Assume that

C(j)f(σ,σ, . . . ,σj+) =C(j)f

for every-cycleσ,σ, . . . ,σj+∈Sm,j= , , . . . ,n– .Then for any x,y,β,τi∈Smand

rear-rangementsπ,whereβ,τiare-cycles,we have

f(ττ· · ·τp) =f(τπ()τπ()· · ·τπ(p)), (.)

f(xβy) =fx()y, (.)

f(β) =f() (.)

for every f ∈KerC(n)_(S

m,H).

Proof First, for any -cycleτi∈Sm,i= , , . . . ,p, and rearrangementπ, it follows from

(.), (.), and (.) that whenp≥n,

f(ττ· · ·τp)

=

p

i= f(τi) +

≤i<i≤p

Cf(τi,τi) +

≤i<i<i≤p

C()f(τi,τi,τi)

+· · ·+ ≤i<i<···<in≤p

C(n–)f(τi,τi, . . . ,τin)

=

p

i=

f(τπ(i)) +

p(p– )  Cf

(), ()+p(p– )(p– )

 C

()_{f(), (), ()}

+· · ·+p(p– )(p– )· · ·(p–n+ )

(n)! C

(n–)_{f(), (), . . . , ()}

=f(τπ()τπ()· · ·τπ(p)),

which gives the case ofp≥nin (.). The case ofp<nis similar to verify. This confirms the proof of (.).

On the other hand, for allx,y,β∈Sm, there exist -cyclesσi,τj,z∈Sm,i= , , . . . ,p,j=

, , . . . ,q, such thatx=σσ· · ·σp,y=ττ· · ·τq, andβ=z()z–. Noting thatz=δδ· · ·δr

for some -cyclesδ,δ, . . . ,δr∈Sm, we obtain

f(xβy) =fσσ· · ·σpδδ· · ·δr()δ–r δ–r–· · ·δ–ττ· · ·τq

=fσσ· · ·σp()δδ· · ·δrδ–r δ–r–ττ· · ·τq

=fx()y

by (.). In particular, takingx=y=ein (.), we get formula (.). This completes the

proof.

According to Lemma , we give the following main result in this section.

Theorem  Assume that(.)holds.Then f is a solution to the equation C(n)_(S

m,H) = 

if and only if there is h∈H such thatnh= and

f(x) = ⎧ ⎨ ⎩

 if x is even,

h if x is odd.

Proof We first deal with the necessity. Letf ∈KerC(n)_(S

We claim that

where

proof of (.). According to (.) and (.), equation (.) becomes

+

k k

(–)k–k(– + )n+–k

h

= ,

which confirms the odd case. With a similar discussion, the proof of the even case is also

obtained.

5 Solutions on finite cyclic groups

LetCm=a|am=e be a cyclic group of ordermwith generatora. In this section, we

study a general solution on the finite cyclic groupCm.

Theorem  Assume that mC(k)_{f(a,a, . . . ,a}

k+

) = ,k= , , . . . ,n– ,and m=n,mf(a) = .

Then f is a solution to the equation C(n)(Cm,H) = if and only if it is given by

fap= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

n–

j=

p(p–)···(p–j) (j+)! C

(j)_{f(a,a, . . . ,a}

j+

) if p≤or p≥n,

p–

j=

p(p–)···(p–j)

(j+)! C(j)f(a,a, . . . ,a) if <p<n,

(.)

where p∈Zand f(a),C(n–)_{f(a,a, . . . ,a)satisfy}

mC(n–)f(a,a, . . . ,a) = , (.)

mf(a) =  if m<n or m>n, (.)

C(m–)f(a,a, . . . ,a) =  if m=n. (.)

Proof Letf :Cm→Hbe a function satisfyingC(n)f = . Then, by (.) we see thatf also

satisfies (.). Now using (.), (ii) of Proposition , and the factam_{=e, we get}

mC(n–)_{f(a,a, . . . ,a) =C}(n–)_fam_{,a, . . . ,a=C}(n–)_{f(e,a, . . . ,a) = ,}

which gives (.). Furthermore, let p= m in (.), according to the assumptions of mC(k)_{f(a,a, . . . ,a) = ,k= , , . . . ,n– ,mf(a) = , andm=n, (.)-(.) are obtained.} This proves (.)-(.) and implies the necessity.

To give out the sufficiency, we claim that (.)-(.) give a well-defined function onCm.

Indeed, whenm>n, it suffices to verify the following four cases: (i)p+m≥nandn–m≤ p≤ orp≥n; (ii)p+m≥nand  <p<n; (iii)p+m≤ andp≤–m< ; (iv)  <p+m<n and –m<p<n–m< .

For case (i), by (.) we have

fap+m–fap

=

n–

j=

(p+m)(p+m– )· · ·(p+m–j)

(j+ )! C

(j)_{f(a,a, . . . ,a)}

–

n–

j=

p(p– )· · ·(p–j) (j+ )! C

=mf(a) +

n–

j=

(p+m)(p+m– )· · ·(p+m–j)

(j+ )! –

p(p– )· · ·(p–j) (j+ )!

×C(j)f(a,a, . . . ,a). (.)

It is easy to see that the coefficient C(j)f(a,a, . . . ,a) is an integer multiple ofm forj= , , . . . ,n– , and therefore, by (.)-(.) and the assumption ofmC(k)_{f(a,a, . . . ,a) = ,} k= , , . . . ,n– , (.) equals .

For case (ii), we compute that

fap+m–fap

=

n–

j=

(p+m)(p+m– )· · ·(p+m–j)

(j+ )! C

(j)_{f(a,a, . . . ,a}

j+ )

–

p–

j=

p(p– )· · ·(p–j) (j+ )! C

(j)_{f(a,a, . . . ,a}

j+ )

=mf(a) +

p–

j=

(p+m)(p+m– )· · ·(p+m–j) (j+ )!

–p(p– )· · ·(p–j) (j+ )!

C(j)f(a, a, . . . ,a

j+ )

+

n–

j=p

(p+m)(p+m– )· · ·(p+m–j)

(j+ )! C

(j)_{f(a,a, . . . ,a}

j+

). (.)

With the same discussion as in case (i), (.) equals . The proofs of the other two cases are similar to case (i).

Whenm<n, it suffices to verify the following five cases: (i)p+m≥nandn–m<p<n; (ii)p+m≥nandp≥n; (iii)p+m≤ andp≤–m; (iv)  <p+m<nand –m<p< ; (v)  <p+m<nand  <p<n–m. Since the proof is similar to the case ofm>n, we omit the details.

Whenm=n, it suffices to verify the following four cases: (i)p+m≥nand  <p<n; (ii)p+m≥nandp≥n; (iii)p+m≤ andp≤–m; (iv)  <p+m<nand –m<p< . Similarly to the cases ofm>nandm<n, the proof ofm=nis omitted.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors are grateful to the referee for his valuable comments and suggestions, especially for pointing out some valuable references. This work is partially supported by general scientific research project of Zhejiang Educational Committee (Y201431986), the National Science Foundation of China (#11301226, #61202109) and Zhejiang Provincial Natural Science Foundation of China under Grant (#LQ13A010017).

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