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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 609862, 10 pages
http://dx.doi.org/10.1155/2012/609862
Research Article

A Linear Homogeneous Partial Differential Equation with Entire Solutions Represented by Laguerre Polynomials

1College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
2Department of Mathematics, Shandong University, Shandong, Jinan 250100, China

Received 11 November 2011; Revised 19 March 2012; Accepted 19 March 2012

Academic Editor: Agacik Zafer

Copyright © 2012 Xin-Li Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study a homogeneous partial differential equation and get its entire solutions represented in convergent series of Laguerre polynomials. Moreover, the formulae of the order and type of the solutions are established.

1. Introduction and Main Results

The existence and behavior of global meromorphic solutions of homogeneous linear partial differential equations of the second order where are polynomials for , have been studied by Hu and Yang [1]. Specially, in [1, 2], they have studied the following cases of  (1.1) and showed that the solutions of (1.2) and (1.3) are closely related to Bessel functions and Bessel polynomials, respectively. Hu and Li [3] studied meromorphic solutions of homogeneous linear partial differential equations of the second order in two independent complex variables: where . Equation (1.4) has a lot of entire solutions on represented by Jacobian polynomials. Global solutions of some first-order partial differential equations (or system) were studied by Berenstein and Li [4], Hu and Yang [5], Hu and Li [6], Li [7], Li and Saleeby [8], and so on.

In this paper, we concentrate on the following partial differential equation (PDE) for a real . We will characterize the entire solutions of (1.5), which are related to Laguerre polynomials. Further, the formulae of the order and type of the solutions are obtained.

It is well known that the Laguerre polynomials are defined by which are solutions of the following ordinary differential equations (ODE): Moreover, Hu [9] pointed out that the generating function of is a solution of the PDE (1.5). Based on the methods from Hu and Yang [2], we get the following results.

Theorem 1.1. The partial differential equation (1.5) has an entire solution on , if and only if has a series expansion such that

If is an entire function on , set we define its order by where

Theorem 1.2. If is defined by (1.9) and (1.10), then where

Valiron [10] showed that each entire solution of a homogeneous linear ODE with polynomial coefficients was of finite order. By studying (1.2) and (1.3), Hu and Yang showed that Valiron’s theorem was not true for general partial differential equations. Here by using Theorems 1.1 and 1.2, we can construct entire solution of (1.5) with arbitrary order .

If , we define the type of by

Theorem 1.3. If is defined by (1.9) and (1.10), and , then the type satisfies

Lindelöf-Pringsheim theorem [11] gave the expression of order and type for one complex variable entire function, and for two variable entire function the formulae of order and type were obtained by Bose and Sharma in [12]. Hu and Yang [2] established an analogue of Lindelöf-Pringsheim theorem for the entire solution of PDE (1.2). But from Theorems 1.2 and 1.3, we find that the analogue theorem for the entire solution of (1.5) is different from the results due to Hu and Yang.

2. An Estimate of Laguerre Polynomials

Before we prove our theorems, we give an upper bound of , which will play an important role in this paper. The following asymptotic properties of can be found in [13]:

(a) holds for in the complex plane cut along the positive real semiaxis; thus, for , we obtain that holds when is large enough.

(b) holds uniformly on compact subsets of , where is the Bessel function and combining with (2.3), for we can deduce that holds when is large enough. Then (2.2) and (2.5) imply where

3. Proof of Theorem 1.1

Assuming that is an entire solution on satisfying (1.5), we have Taylor expansion where Hence is an entire solution of (1.7).

By the method of Frobenius (see [14]), we can get a second independent solution of (1.7) which is where are constants.

So there exist and satisfying Because of the singularity of at , we obtain . That shows

Now we need to estimate the terms of . Since is an entire function, we have Since we easily get

Conversely, the relations (1.7), (1.9), and (1.10) imply that holds for all . Since (2.6) implies we have Combining (1.10), (3.10) with (3.12), we can get that is obviously an entire solution of (1.5) on .

4. Proof of Theorem 1.2

Firstly, we prove . If , the result is trivial. Now we assume and prove for any . The relation (1.15) implies that there exists a sequence such that By using Cauchy’s inequality of holomorphic functions, we have together with the formula of the coefficients of the Taylor expansion we obtain . Since , we have then Putting , we have which means . Then we can get .

Next, we will prove . Set . The result is easy for ; then we assume . For any , (1.15) implies that there exists , when , we have where . For any , there exists such that when , combining with (2.6) and (4.7), we get where is a constant but not necessary to be the same every time.

Set , which means that for . Further set , which yields that when , Obviously, we can choose such that for . Then We also have Therefore, when , we have which means . Hence follows by letting .

5. Proof of Theorem 1.3

Set At first, we prove . The result is trivial for , we assume and take with , set Equation (5.1) implies that there exists a sequence satisfying combining with (4.4), we can deduce that Taking , we get , which yields , so .

Next, we prove . We may assume . Equation (5.1) implies that for any , there exists , such that when , For any , we choose such that when , combining with (2.6), we have

Set , when , we deduce . Set , it is obvious that for . Since , there exists such that when , Then We note that for , , , then we have This shows

Therefore when , Together with and the definition of type, we can get , which yields by letting .

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. The first author was partially supported by Natural Science Foundation of China (11001057), and the third author was partially supported by Natural Science Foundation of Shandong Province.

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