Open Access

Some results on q-difference equations

Advances in Difference Equations20122012:191

https://doi.org/10.1186/1687-1847-2012-191

Received: 16 June 2012

Accepted: 24 October 2012

Published: 5 November 2012

Abstract

In this paper, we consider the q-difference analogue of the Clunie theorem. We obtain there is no zero-order entire solution of f n ( z ) + ( q f ( z ) ) n = 1 when n 2 ; there is no zero-order transcendental entire solution of f n ( z ) + P ( z ) ( q f ( z ) ) m = Q ( z ) when n > m 0 ; and the equation f n + P ( z ) q f ( z ) = h ( z ) has at most one zero-order transcendental entire solution f if f is not the solution of q f ( z ) = 0 , when n 4 .

MSC:30D35, 30D30, 39A13, 39B12.

Keywords

uniqueness q-shift q-difference equations entire functions zero order Nevanlinna theory

1 Introduction and main results

It is well known that Clunie’s theorem (see [1], Lemma 1; also see [2], p.39, Lemma 2.4.1) is a useful tool in studying complex differential equations. It states that Q n ( f ) is a polynomial of total degree n at most in the meromorphic function f and its derivatives having meromorphic functions as coefficients. If T ( r ) is the maximum of the characteristics of the coefficients, then
1 2 π | f | > 1 log + | f n Q n ( f ) | d φ = O ( log r + log T ( r , f ) + T ( r ) ) n.e. as  r .

Later, Clunie’s theorem has been improved into many forms (see [2], pp.39-44) which are valuable tools for studying meromorphic solutions of Painlevé and other non-linear differential equations; see, e.g., [2].

In 2007, Laine and Yang [3] obtained a discrete version of Clunie’s theorem.

Theorem A Let f be a transcendental meromorphic solution of finite-order ρ of a difference equation of the form
U ( z , f ) P ( z , f ) = Q ( z , f ) ,
where U ( z , f ) , P ( z , f ) , and Q ( z , f ) are difference polynomials such that the total degree deg U ( z , f ) = n in f ( z ) and its shifts, and deg Q ( z , f ) n . Moreover, we assume that U ( z , f ) contains just one term of maximal total degree in f ( z ) and its shifts. Then for each ε > 0 ,
m ( r , P ( z , f ) ) = O ( r ρ 1 + ε ) + S ( r , f )

possibly outside of an exceptional set of finite logarithmic measure.

Now let us introduce some notation. Let c j C for j = 1 , , n , and let I be a finite set of multi-indexes λ = ( λ 0 , , λ n ) . A difference polynomial of a meromorphic function w ( z ) is defined as
P ( z , w ) = P ( z , w ( z ) , w ( z + c 1 ) , , w ( z + c n ) ) = λ I a λ ( z ) w ( z ) λ 0 w ( z + c 1 ) λ 1 w ( z + c n ) λ n ,
(1.1)
where the coefficients a λ ( z ) are small with respect to w ( z ) in the sense that T ( r , a λ ) = o ( T ( r , w ) ) as r tends to infinity outside of an exceptional set E of finite logarithmic measure
lim r E [ 1 , r ) d t t < .
The total degree of P ( z , w ) in w ( z ) and in the shifts of w ( z ) is denoted by deg w ( P ) , and the order of a zero of P ( z , x 0 , x 1 , , x n ) , as a function of x 0 at x 0 = 0 , is denoted by ord 0 ( P ) ; see, e.g., [4]. Moreover, the weight of a difference polynomial (1.1) is defined by
K ( P ) = max λ I { j = 1 n λ j } ,

where λ and I are the same as in (1.1) above. The difference polynomial P ( z , w ) is said to be homogeneous with respect to w ( z ) if the degree d λ = λ 0 + + λ n of each term in the sum (1.1) is non-zero and the same for all λ I .

Recently, Korhonen obtained a new Clunie-type theorem in [4].

Theorem B Let w ( z ) be a finite-order meromorphic solution of
H ( z , w ) P ( z , w ) = Q ( z , w ) ,
where P ( z , w ) is a homogeneous difference polynomial with meromorphic coefficients, and H ( z , w ) and Q ( z , w ) are polynomials in w ( z ) with meromorphic coefficients having no common factors. If
max { deg w ( H ) , deg w ( Q ) deg w ( P ) } > min { deg w ( P ) , ord 0 ( Q ) } ord 0 ( P ) ,

then N ( r , w ) S ( r , w ) .

Theorem C Let w ( z ) be a finite-order meromorphic solution of
H ( z , w ) P ( z , w ) = Q ( z , w ) ,
where P ( z , w ) is a homogeneous difference polynomial with meromorphic coefficients, and H ( z , w ) and Q ( z , w ) are polynomials in w ( z ) with meromorphic coefficients having no common factors. If
2 K ( P ) max { deg w ( Q ) , deg w ( H ) + deg w ( P ) } min { deg w ( P ) , ord 0 ( Q ) } ,
then for any δ ( 0 , 1 ) ,
m ( r , w ) = o ( T ( r , w ) r δ ) + O ( T ( r ) ) ,

where r goes to infinity outside of an exceptional set of finite logarithmic measure, and T ( r ) is the maximum of the Nevanlinna characteristics of the coefficients of P ( z , w ) , Q ( z , w ) , and H ( z , w ) .

The non-autonomous Schröder q-difference equation
f ( q z ) = R ( z , f ( z ) ) ,
(1.2)

where the right-hand side is rational in both arguments, has been widely studied during the last decades; see, e.g., [58]. Gundersen et al. [9] considered the order of growth of meromorphic solutions of (1.2), from which a q-difference analogue of the classical Malmquist theorem [10] is given: if the q-difference equation (1.2) admits a meromorphic solution of order zero, then (1.2) reduces to a q-difference Riccati equation, i.e., deg f R = 1 .

Bergweiler et al. [11] treated the functional equation
j = 0 n a j ( z ) f ( c j z ) = Q ( z ) ,
(1.3)

where 0 < | c | < 1 is a complex number, a j ( z ) ( j = 0 , 1 , , n ), and Q ( z ) are rational functions with a 0 ( z ) 0 , a 1 ( z ) 1 . They concluded that all meromorphic solutions of (1.3) satisfy T ( r , f ) = O ( ( log r ) 2 ) . This implies that all meromorphic solutions of (1.3) are of zero order of growth.

Let us recall Δ c f ( z ) of a meromorphic function in the whole plane is given by
Δ c f ( z ) = f ( z + c ) f ( z ) ,
while
q f ( z ) = f ( q z ) f ( z )
denotes q f ( z ) of a meromorphic function in the whole plane , where c C { 0 } and q C { 0 , 1 } . The upper logarithmic density of E is defined by
log dens ( E ) ¯ = lim sup r E [ 1 , r ] d t t log r .

In particular, we denote by S q ( r , f ) any quantity satisfying S q ( r , f ) = o ( T ( r , f ) ) for all r outside of a set of upper logarithmic density 0 on the set of logarithmic density 1.

In 2009, Liu [12] proved the following the result.

Theorem D There is no non-constant entire solution with finite order of the non-linear difference equation
f 2 ( z ) + ( Δ c f ( z ) ) 2 = 1 .
(1.4)

It is well known that f n ( z ) + g n ( z ) = 1 has no entire solutions when n 3 (see [13], Theorem 3), and from Theorem D, we can say there is no non-constant entire solutions with finite order of the equation f n ( z ) + ( Δ c f ( z ) ) n = 1 , when n 2 .

In this paper, we replace Δ c f ( z ) by q f ( z ) and get the following result.

Theorem 1 There is no non-constant entire solution with zero order of the non-linear q-difference equation
f n ( z ) + ( q f ( z ) ) n = 1 ,
(1.5)

when n 2 .

Theorem 2 Let P ( z ) and Q ( z ) be polynomials, and let n and m be integers satisfying n > m 0 . Then there is no non-constant entire transcendental solution with zero order of the non-linear q-difference equation
f n ( z ) + P ( z ) ( q f ( z ) ) m = Q ( z ) .
(1.6)

In 2010, Yang and Laine [14] got the following result.

Theorem E Let n 4 be an integer, M ( z , f ) be a linear difference polynomial of f, not vanishing identically, and h be a meromorphic function of finite order. Then the difference equation
f n + M ( z , f ) = h

possesses at most one admissible transcendental entire solution of finite order such that all coefficients of M ( z , f ) are small functions of f. If such a solution f exists, then f is of the same order as h.

In this paper, we replace difference polynomial by q-difference polynomial and get the following result.

Theorem 3 Let n 4 be an integer, M ˜ ( r , f ) be a linear q-difference polynomial of f, not vanishing identically, and h ( z ) be a meromorphic function. Suppose f ( z ) is the solution of q-difference equation
f n + N ( r , f ) = h ( z ) .
(1.7)

If f ( z ) is not the solution of M ˜ ( r , f ) = 0 , then equation (1.7) possesses at most one transcendental entire solution of zero order.

2 Auxiliary results

The following auxiliary results will be instrumental in proving the theorems.

Lemma 1 ([5], Theorem 1.2)

Let f ( z ) be a non-constant zero-order meromorphic function and q C { 0 } . Then
m ( r , f ( q z ) f ( z ) ) = S q ( r , f ) .

Lemma 2 ([5], Theorem 2.1)

Let f ( z ) be a non-constant zero-order meromorphic solution of
f n ( z ) P ( z , f ) = Q ( z , f ) ,
where P ( z , f ) and Q ( z , f ) are q-difference polynomials in f ( z ) . If the degree of Q ( z , f ) as a polynomial in f ( z ) and its q-shifts is at most n, then
m ( r , P ( z , f ) ) = S q ( r , f ) .

Lemma 3 ([15], Theorem 1.1)

Let f ( z ) be a non-constant zero-order meromorphic function and q C { 0 } . Then
T ( r , f ( q z ) ) = T ( r , f ) + S q ( r , f ) .

Lemma 4 ([15], Theorem 1.3)

Let f ( z ) be a non-constant zero-order meromorphic function and q C { 0 } . Then
N ( r , f ( q z ) ) = N ( r , f ) + S q ( r , f ) .

Lemma 5 ([16], Lemma 4)

If T : R + R + is a piecewise continuous increasing function such that
lim r log T ( r ) log r = 0 ,
then the set
E : = { r : T ( C 1 r ) C 2 T ( r ) }

has logarithmic density 0 for all C 1 > 1 and C 2 > 1 .

Lemma 6 Let f ( z ) be a zero-order entire function, q C { 0 } , and a be a non-zero constant. If f ( z ) and q f ( z ) share the set { a , a } CM, then f ( z ) is a constant.

Proof Since f ( z ) is an entire function of zero order, and f ( z ) and q f ( z ) share the set { a , a } CM, it is immediate to conclude that
( f ( z ) a ) ( f ( z ) + a ) ( q f ( z ) + a ) ( q f ( z ) a ) k 2 ,
(2.1)

where k is a constant.

If k 2 1 , let
h 1 ( z ) : = f ( z ) + k q f ( z ) ,
(2.2)
and
h 2 ( z ) : = f ( z ) k q f ( z ) .
(2.3)
Then h 1 ( z ) and h 2 ( z ) are entire functions, and
f ( z ) = h 1 ( z ) + h 2 ( z ) 2 , q f ( z ) = h 1 ( z ) h 2 ( z ) 2 k .
(2.4)
From (2.1)-(2.3), we have
h 1 ( z ) h 2 ( z ) = f 2 ( z ) k 2 ( q f ( z ) ) 2 = ( 1 k 2 ) a 2 .
(2.5)
From (2.2), (2.3), and (2.5), we obtain
N ( r , 1 h 1 ) = N ( r , 1 h 2 ) = 0 .
(2.6)

Since h 1 and h 2 with zero order have no zeros and no poles, both h 1 and h 2 are constants. (2.4) implies that f ( z ) is a constant.

If k 2 = 1 , from (2.1) we get f ( z ) = q f ( z ) . According to Lemma 3, it implies that f ( z ) must be a constant. □

3 Clunie theorem for q-difference

Let us consider the q-difference polynomial case. Let d j C for j = 1 , , n , and let I q be a finite set of multi-indexes γ = ( γ 0 , , γ n ) . A difference polynomial of a meromorphic function w ( z ) is defined as
P ( z , w ) = P ( z , w ( q z ) , w ( q 2 z ) , , w ( q n z ) ) = γ I q a γ ( z ) w ( z ) γ 0 w ( q z ) γ 1 w ( q n z ) γ n ,
(3.1)
where the coefficients a γ ( z ) are small with respect to w ( z ) in the sense that T ( r , a γ ) = o ( T ( r , w ) ) as r tends to infinity outside of an exceptional set E of finite logarithmic measure
lim r E [ 1 , r ) d t t < .
The total degree of P ( z , w ) in w ( z ) and in the q-shifts of w ( z ) is denoted by deg w q ( P ) , and the order of a zero of P ( z , x 0 , x 1 , , x n ) , as a function of x 0 at x 0 = 0 , is denoted by ord 0 q ( P ) ; see, e.g., [4]. Moreover, the weight of a difference polynomial (1.1) is defined by
K q ( P ) = max γ I q { j = 1 n γ j } ,

where γ and I q are the same as in (3.1) above. The difference polynomial P ( z , w ) is said to be homogeneous with respect to w ( z ) , if the degree d γ = γ 0 + + γ n of each term in the sum (1.1) is non-zero and the same for all γ I q .

In this paper, we will obtain the new Clunie theorem for q-difference polynomials.

Theorem 4 Let w ( z ) be a zero-order meromorphic solution of
H ( z , w ) P ( z , w ) = Q ( z , w ) ,
where P ( z , w ) is a homogeneous q-difference polynomial with polynomial coefficients, and H ( z , w ) and Q ( z , w ) are polynomials in w ( z ) with polynomial coefficients having no common factors. If
max { deg w q ( H ) , deg w q ( Q ) deg w q ( P ) } > min { deg w q ( P ) , ord 0 q ( Q ) } ord 0 q ( P ) ,

then N ( r , w ) S q ( r , w ) .

Proof Since P ( r , w ) is homogeneous, by Lemma 1 it follows that
m ( r , P ( r , w ) w deg w q ( P ) ) = S q ( r , w ) .
(3.2)
Moreover, Mohon’ko’s theorem (see [17], Theorem 1.13) implies that
T ( r , P ( r , w ) w deg w q ( P ) ) = d w T ( r , w ) + O ( log r ) ,
(3.3)
where
d m = max { deg w q ( Q ) , deg w q ( H ) + deg w q ( P ) } min { deg w q ( P ) , ord 0 q ( Q ) } .
(3.4)
According to (3.2), (3.3), (3.4), and the assumption of Theorem 4, it follows that
N ( r , P ( r , w ) w deg w q ( P ) ) ( 1 + deg w q ( P ) ord 0 q ( P ) ) T ( r , w ) + S q ( r , w ) .
(3.5)
This contradicts the assertion of Theorem 4 that N ( r , w ) = S q ( r , w ) . Let us denote q max = max { | q j | , ( j = 1 , , n ) } , by Lemma 5 we will obtain that
N ( r , P ( r , w ) w ord 0 q ( P ) ) ( deg w q ( P ) ord 0 q ( P ) ) N ( q max r , w ) ( deg w q ( P ) ord 0 q ( P ) + o ( 1 ) ) N ( r , w ) + S q ( r , w ) = S q ( r , w ) .
Therefore,
N ( r , P ( r , w ) w deg w q ( P ) ) N ( r , P ( r , w ) w ord 0 q ( P ) ) + N ( r , 1 w deg w q ( P ) ord 0 q ( P ) ) = N ( r , 1 w deg w q ( P ) ord 0 q ( P ) ) + S q ( r , w ) T ( r , 1 w deg w q ( P ) ord 0 q ( P ) ) + S q ( r , w ) = ( deg w q ( P ) ord 0 q ( P ) ) T ( r , w ) + S q ( r , w ) ,

which is a contradiction to (3.5). We can conclude that N ( r , w ) S q ( r , w ) . □

Theorem 5 Let w ( z ) be a zero-order meromorphic solution of
H ( z , w ) P ( z , w ) = Q ( z , w ) ,
where P ( z , w ) is a homogeneous q-difference polynomial with polynomial coefficients, and H ( z , w ) and Q ( z , w ) are polynomials in w ( z ) with polynomial coefficients having no common factors. If
2 K q ( P ) max { deg w q ( Q ) , deg w q ( H ) + deg w q ( P ) } min { deg w q ( P ) , ord 0 q ( Q ) } ,
then
m ( r , w ) = S q ( r , w ) .
Proof On the one hand, (3.2) and (3.5) imply that
N ( r , P ( r , w ) w deg w q ( P ) ) = d w T ( r , w ) + S q ( r , w ) .
(3.6)
On the other hand, by Lemma 4, we can obtain that
N ( r , P ( r , w ) w deg w q ( P ) ) K q ( P ) ( 2 T ( r , w ) m ( r , w ) ) + S q ( r , w ) .
(3.7)
(3.6) and (3.7) show that
m ( r , w ) = S q ( r , w ) .

 □

4 Proof of Theorem 1

If n 3 , there are no non-constant entire solutions of equation (1.5) according to [13]. Let us consider n = 2 , that is,
f 2 ( z ) + ( Δ q f ( z ) ) 2 = 1 .
(4.1)

From (4.1), we get f ( z ) and q f ( z ) share the set { 2 2 , 2 2 } CM. From Lemma 6, we obtain that f ( z ) is a constant.

5 Proof of Theorem 2

Suppose that f is a transcendental entire solution of equation (1.6) with zero order. If q f ( z ) 0 , then f n ( z ) = Q ( z ) , and the conclusion holds. If q f ( z ) 0 , we have
f n 1 f = Q ( z ) P ( z ) ( Δ q f ( z ) ) f m f m .
From Lemma 1, Lemma 2, and the condition n > m , we have
T ( r , f ) = m ( r , f ) = S q ( r , f ) ,

which is impossible.

6 Proof of Theorem 3

Assume now, contrary to the assertion, that f and g, which are not the solutions of M ˜ ( r , f ) = 0 and M ˜ ( r , g ) = 0 , are two distinct zero-order transcendental entire solutions of (1.7), then we can write
f n + M ˜ ( r , f ) = g n + M ˜ ( r , g ) .
(6.1)
From (6.1), we obtain
f n g n = M ˜ ( r , g ) M ˜ ( r , f ) .
(6.2)
Therefore, we have
F : = f n ( z ) g n ( z ) f ( z ) g ( z ) = j = 1 n 1 ( f η j g ) = M ˜ ( r , f ) M ˜ ( r , g ) f ( z ) g ( z )
(6.3)
is an entire function, and η 1 , , η n 1 are distinct roots ≠1 of the equation z n = 1 . Hence, N ( r , M ˜ ( r , f ) M ˜ ( r , g ) f ( z ) g ( z ) ) = 0 . From Lemma 1, we get
T ( r , M ˜ ( r , f ) M ˜ ( r , g ) f ( z ) g ( z ) ) = S q ( r , f g ) .
(6.4)
If f g is not a constant, (6.3) implies that
( n 1 ) T ( r , f g ) = T ( r , g n 1 ) + S q ( r , f g ) .
(6.5)
Thus,
j = 1 n 1 N ( r , 1 f g η j ) = N ( r , f g M ˜ ( r , f ) M ˜ ( r , g ) ) T ( r , f g M ˜ ( r , f ) M ˜ ( r , g ) ) = S q ( r , f g ) .
From the second fundamental theorem, we have
( n 3 ) T ( r , f g ) j = 1 n 1 N ( r , 1 f g η j ) + O ( log r ) = S q ( r , f g ) .
(6.6)
From (6.6), (6.5), and f, g are zero-order entire functions, we get a contradiction. Therefore, f g must be a constant. If f / g = k η j , k is a constant, then from (6.1) we have
( k n 1 ) g n = M ˜ ( r , ( 1 k ) g ) .
(6.7)
From Lemma 2 and (6.7), we get a contradiction. Thus, f / g = η j for some j = 1 , , n 1 and
f ( z ) = η j g ( z ) .
(6.8)
This implies that
f n = g n
(6.9)
and
M ˜ ( r , f ) = η j M ˜ ( r , g ) .
(6.10)
From (6.9) and (6.2), we obtain
M ˜ ( r , f ) = M ˜ ( r , g ) .
(6.11)

From (6.9) and (6.2), it is easy to get M ˜ ( r , f ) = 0 and M ˜ ( r , g ) = 0 , which is impossible.

Declarations

Authors’ Affiliations

(1)
College of Computer Science and Technology, Taiyuan University of Technology
(2)
Shandong Transport Vocational College
(3)
Shanxi Taiyuan Tideflow Electronic Technology Co., Ltd.

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