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Some results on q-difference equations
Advances in Difference Equations volume 2012, Article number: 191 (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 when ; there is no zero-order transcendental entire solution of when ; and the equation has at most one zero-order transcendental entire solution f if f is not the solution of , when .
MSC:30D35, 30D30, 39A13, 39B12.
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 is a polynomial of total degree n at most in the meromorphic function f and its derivatives having meromorphic functions as coefficients. If is the maximum of the characteristics of the coefficients, then
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
where , , and are difference polynomials such that the total degree in and its shifts, and . Moreover, we assume that contains just one term of maximal total degree in and its shifts. Then for each ,
possibly outside of an exceptional set of finite logarithmic measure.
Now let us introduce some notation. Let for , and let I be a finite set of multi-indexes . A difference polynomial of a meromorphic function is defined as
where the coefficients are small with respect to in the sense that as r tends to infinity outside of an exceptional set E of finite logarithmic measure
The total degree of in and in the shifts of is denoted by , and the order of a zero of , as a function of at , is denoted by ; see, e.g., [4]. Moreover, the weight of a difference polynomial (1.1) is defined by
where λ and I are the same as in (1.1) above. The difference polynomial is said to be homogeneous with respect to if the degree of each term in the sum (1.1) is non-zero and the same for all .
Recently, Korhonen obtained a new Clunie-type theorem in [4].
Theorem B Let be a finite-order meromorphic solution of
where is a homogeneous difference polynomial with meromorphic coefficients, and and are polynomials in with meromorphic coefficients having no common factors. If
then .
Theorem C Let be a finite-order meromorphic solution of
where is a homogeneous difference polynomial with meromorphic coefficients, and and are polynomials in with meromorphic coefficients having no common factors. If
then for any ,
where r goes to infinity outside of an exceptional set of finite logarithmic measure, and is the maximum of the Nevanlinna characteristics of the coefficients of , , and .
The non-autonomous Schröder q-difference equation
where the right-hand side is rational in both arguments, has been widely studied during the last decades; see, e.g., [5–8]. 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., .
Bergweiler et al. [11] treated the functional equation
where is a complex number, (), and are rational functions with , . They concluded that all meromorphic solutions of (1.3) satisfy . This implies that all meromorphic solutions of (1.3) are of zero order of growth.
Let us recall of a meromorphic function in the whole plane ℂ is given by
while
denotes of a meromorphic function in the whole plane ℂ, where and . The upper logarithmic density of E is defined by
In particular, we denote by any quantity satisfying 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
It is well known that has no entire solutions when (see [13], Theorem 3), and from Theorem D, we can say there is no non-constant entire solutions with finite order of the equation , when .
In this paper, we replace by and get the following result.
Theorem 1 There is no non-constant entire solution with zero order of the non-linear q-difference equation
when .
Theorem 2 Let and be polynomials, and let n and m be integers satisfying . Then there is no non-constant entire transcendental solution with zero order of the non-linear q-difference equation
In 2010, Yang and Laine [14] got the following result.
Theorem E Let be an integer, be a linear difference polynomial of f, not vanishing identically, and h be a meromorphic function of finite order. Then the difference equation
possesses at most one admissible transcendental entire solution of finite order such that all coefficients of 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 be an integer, be a linear q-difference polynomial of f, not vanishing identically, and be a meromorphic function. Suppose is the solution of q-difference equation
If is not the solution of , 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 be a non-constant zero-order meromorphic function and . Then
Lemma 2 ([5], Theorem 2.1)
Let be a non-constant zero-order meromorphic solution of
where and are q-difference polynomials in . If the degree of as a polynomial in and its q-shifts is at most n, then
Lemma 3 ([15], Theorem 1.1)
Let be a non-constant zero-order meromorphic function and . Then
Lemma 4 ([15], Theorem 1.3)
Let be a non-constant zero-order meromorphic function and . Then
Lemma 5 ([16], Lemma 4)
If is a piecewise continuous increasing function such that
then the set
has logarithmic density 0 for all and .
Lemma 6 Let be a zero-order entire function, , and a be a non-zero constant. If and share the set CM, then is a constant.
Proof Since is an entire function of zero order, and and share the set CM, it is immediate to conclude that
where k is a constant.
If , let
and
Then and are entire functions, and
From (2.1)-(2.3), we have
From (2.2), (2.3), and (2.5), we obtain
Since and with zero order have no zeros and no poles, both and are constants. (2.4) implies that is a constant.
If , from (2.1) we get . According to Lemma 3, it implies that must be a constant. □
3 Clunie theorem for q-difference
Let us consider the q-difference polynomial case. Let for , and let be a finite set of multi-indexes . A difference polynomial of a meromorphic function is defined as
where the coefficients are small with respect to in the sense that as r tends to infinity outside of an exceptional set E of finite logarithmic measure
The total degree of in and in the q-shifts of is denoted by , and the order of a zero of , as a function of at , is denoted by ; see, e.g., [4]. Moreover, the weight of a difference polynomial (1.1) is defined by
where γ and are the same as in (3.1) above. The difference polynomial is said to be homogeneous with respect to , if the degree of each term in the sum (1.1) is non-zero and the same for all .
In this paper, we will obtain the new Clunie theorem for q-difference polynomials.
Theorem 4 Let be a zero-order meromorphic solution of
where is a homogeneous q-difference polynomial with polynomial coefficients, and and are polynomials in with polynomial coefficients having no common factors. If
then .
Proof Since is homogeneous, by Lemma 1 it follows that
Moreover, Mohon’ko’s theorem (see [17], Theorem 1.13) implies that
where
According to (3.2), (3.3), (3.4), and the assumption of Theorem 4, it follows that
This contradicts the assertion of Theorem 4 that . Let us denote , by Lemma 5 we will obtain that
Therefore,
which is a contradiction to (3.5). We can conclude that . □
Theorem 5 Let be a zero-order meromorphic solution of
where is a homogeneous q-difference polynomial with polynomial coefficients, and and are polynomials in with polynomial coefficients having no common factors. If
then
Proof On the one hand, (3.2) and (3.5) imply that
On the other hand, by Lemma 4, we can obtain that
(3.6) and (3.7) show that
□
5 Proof of Theorem 2
Suppose that f is a transcendental entire solution of equation (1.6) with zero order. If , then , and the conclusion holds. If , we have
From Lemma 1, Lemma 2, and the condition , we have
which is impossible.
6 Proof of Theorem 3
Assume now, contrary to the assertion, that f and g, which are not the solutions of and , are two distinct zero-order transcendental entire solutions of (1.7), then we can write
From (6.1), we obtain
Therefore, we have
is an entire function, and are distinct roots ≠1 of the equation . Hence, . From Lemma 1, we get
If is not a constant, (6.3) implies that
Thus,
From the second fundamental theorem, we have
From (6.6), (6.5), and f, g are zero-order entire functions, we get a contradiction. Therefore, must be a constant. If , k is a constant, then from (6.1) we have
From Lemma 2 and (6.7), we get a contradiction. Thus, for some and
This implies that
and
From (6.9) and (6.2), we obtain
From (6.9) and (6.2), it is easy to get and , which is impossible.
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The first author discovered some essential ideas for the proof of this paper, and made the actual writing. The fours discussed the paper together. The other authors checked the proofs of the paper. All authors read and approved the final manuscript.
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Zhang, J., Wang, G., Chen, J. et al. Some results on q-difference equations. Adv Differ Equ 2012, 191 (2012). https://doi.org/10.1186/1687-1847-2012-191
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DOI: https://doi.org/10.1186/1687-1847-2012-191