Some results on q-difference equations
© Zhang et al.; licensee Springer 2012
Received: 16 June 2012
Accepted: 24 October 2012
Published: 5 November 2012
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.
Keywordsuniqueness q-shift q-difference equations entire functions zero order Nevanlinna theory
1 Introduction and main results
Later, Clunie’s theorem has been improved into many forms (see , pp.39-44) which are valuable tools for studying meromorphic solutions of Painlevé and other non-linear differential equations; see, e.g., .
In 2007, Laine and Yang  obtained a discrete version of Clunie’s theorem.
possibly outside of an exceptional set of finite logarithmic measure.
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 .
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 .
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.  considered the order of growth of meromorphic solutions of (1.2), from which a q-difference analogue of the classical Malmquist theorem  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., .
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.
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  proved the following the result.
It is well known that has no entire solutions when (see , 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.
In 2010, Yang and Laine  got the following result.
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.
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 (, Theorem 1.2)
Lemma 2 (, Theorem 2.1)
Lemma 3 (, Theorem 1.1)
Lemma 4 (, Theorem 1.3)
Lemma 5 (, Lemma 4)
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.
where k is a constant.
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
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.
which is a contradiction to (3.5). We can conclude that . □
4 Proof of Theorem 1
5 Proof of Theorem 2
which is impossible.
6 Proof of Theorem 3
From (6.9) and (6.2), it is easy to get and , which is impossible.
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