Entire solutions of certain type of difference equations
© Liu et al.; licensee Springer. 2014
Received: 23 September 2013
Accepted: 24 January 2014
Published: 11 February 2014
In this paper, we shall study the conditions regarding the existence of transcendental entire solutions of certain type of difference equations. Our results are either supplements to some results obtained recently, or are relating to the conjecture raised in Yang and Laine (Proc. Jpn. Acad., Ser. A, Math. Sci. 86:10-14, 2010). Finally, two relevant conjectures are posed for further studies.
MSC:39B32, 34M05, 30D35.
1 Introduction, notations and main results
Let f denote a transcendental entire or meromorphic function. Assuming the reader is familiar with the basics of Nevanlinna’s value distribution theory, we shall adopt the standard notations associated with the theory, such as the characteristic function , the counting function of the poles , and the proximity function (see, e.g., [1, 2]).
Among many interesting applications of the Nevanlinna theory, there are studies on the growth and existence of entire or meromorphic solutions of various types of non-linear differential equations, and one can find prototypes for such equations, e.g., in [3–5] and . Recently, the Nevanlinna theory has been applied to study types of non-linear difference equations, see, e.g., [7, 8]. Now, we shall utilize Clunie type of theorems for difference-differential polynomials to study some non-linear difference equations of more general forms and to obtain some improvements of or supplements to [7, 9], and .
Notations Given a meromorphic function f, recall that is a small function with respect to f, if , where denotes any quantity satisfying as , possibly outside a set of r of finite linear measure. For a constant , is called a shift of f. As for a difference product, we mean a difference monomial of type , where are complex constants and are natural numbers.
Definition 1.1 A difference polynomial, respectively, a difference-differential polynomial, in f is a finite sum of difference products of f and its shifts, respectively, of products of f, derivatives of f and of their shifts, with all the coefficients of these monomials being small functions of f.
where () are small functions of f, and (; ) are complex constants, and (; ) are natural numbers.
Group together similar terms of , if necessary. In the following, we assume that no two terms of are similar and that ().
For the sake of simplicity, we let , ().
Yang and Laine  considered the following difference equation and proved it.
where is a non-constant polynomial and are nonzero constants, does not admit entire solutions of finite order. If is a nonzero constant, then the above equation possesses three distinct entire solutions of finite order, provided that and for a nonzero integer n.
Now, we shall substitute by in Theorem A and prove the following results.
, and , ,
, and , ,
where , are constants satisfying , .
has no transcendental entire solutions of finite order provided that , where k is an integer.
By some further analysis, we can derive the following result.
where is a non-constant polynomial and are nonzero constants, does not admit entire solutions of finite order. If is a nonzero constant, then (1.2) possesses solutions of the form , , provided that , n is an odd number, or , .
a finite order entire solution is .
And has the entire solution .
where q is a nonzero constant, possesses solutions of the form , if and only if .
Theorem B (, Theorem 2.4)
has no transcendental entire solutions of finite order.
We shall modify the equation in Theorem B above and derive the following result.
has no transcendental entire solutions of hyper-order .
In order to prove our conclusions, we need some lemmas.
Lemma 2.1 ()
possibly outside of an exceptional set of finite logarithmic measure.
Remark 2.1 The following result is a Clunie type lemma  for the difference-differential polynomials of a meromorphic function f. It can be proved by applying Lemma 2.1 with a similar reasoning as in  and stated as follows.
Let be a meromorphic function of finite order, and let , be two difference-differential polynomials of f. If holds and if the total degree of in f and its derivatives and their shifts is ≤n, then .
Lemma 2.2 ()
Suppose that m, n are positive integers satisfying . Then there exist no transcendental entire solutions f and g satisfying the equation , with a, b being small functions of f and g, respectively.
Lemma 2.3 ()
Assume that is a nonzero constant, α is a non-constant meromorphic function. Then the differential equation has no transcendental meromorphic solutions satisfying .
Let f be a transcendental meromorphic function of finite order ρ, then for any complex numbers , and for each , .
outside of a possible exceptional set with finite logarithmic measure.
We also know that Remark 2.2 has been improved by Halburd et al. . They proved that (2.2) is also true when f is a meromorphic function of hyper-order .
3 Proof of Theorem 1.1
cannot hold simultaneously.
where B is a constant.
If , by Lemma 2.1, we get , which is absurd. So , and , , .
If , by (3.3) and using similar arguments as above, we can derive , , .
This completes the proof of Theorem 1.1.
Remark 3.1 Suggested by the referee, one can also derive the conclusions of Theorem 1.1 when .
where . This equation has two constant solutions, , . By Corollary 5.2 in the paper by Bank et al. , all other meromorphic solutions are of infinite order. And from this one can obtain the two entire solutions of exponential type.
4 Proof of Theorem 1.2
where is a difference-differential polynomial of f, and its total degree at most 4.
where , , , , .
where , , .
Now we distinguish three cases to discuss.
and , where , are constants. Therefore n is odd and .
Case 2. Assume that ; by a similar method as Case 1, we get or .
If , combining with , we find , which is a contradiction.
If , substituting this expression of b into , we have .
Case 3. While , using a similar way as above, we get or . If , we can get a contradiction, hence , and .
This completes the proof of Theorem 1.2.
5 Proof of Theorem 1.3
For , , we get a contradiction, thus (1.3) has no transcendental entire solutions of hyper-order .
This also completes the proof of Theorem 1.3.
6 Final remark and conjectures
The current Clunie types of theorems regarding difference-differential polynomials are mainly useful or effective to deal with problems relating to entire or meromorphic solutions of finite order for certain types of difference-differential equations. Thus, it is very natural for us to pose the following two conjectures, for further studies.
Conjecture 6.1 There are no entire solutions of infinite order for any equations (1.1), (1.2), and (1.3).
have no entire solutions of infinite order, where is a difference polynomial, , are polynomials, and are integers, and , , () are constants.
The authors would like to thank the referee for his/her reading of the original version of the manuscript with valuable suggestions and comments.
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