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

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 order𝑎₀𝜕²𝑢𝜕𝑡²+2𝑎₁𝜕²𝑢𝜕𝑡𝜕𝑧+𝑎₂𝜕²𝑢𝜕𝑧²+𝑎₃𝜕𝑢𝜕𝑡+𝑎₄𝜕𝑢𝜕𝑧+𝑎₆𝑢=0, (1.1) where 𝑎_𝑘=𝑎_𝑘(𝑡,𝑧) are polynomials for (𝑡,𝑧)∈ℂ², have been studied by Hu and Yang [1]. Specially, in [1, 2], they have studied the following cases of  (1.1)𝑡²𝜕²𝑢𝜕𝑡²−𝑧²𝜕²𝑢𝜕𝑧²+(2𝑡+2)𝜕𝑢𝜕𝑡−2𝑧𝜕𝑢𝑡𝜕𝑧=0,₂𝜕²𝑢𝜕𝑡²−𝑧²𝜕²𝑢𝜕𝑧²+𝑡𝜕𝑢𝜕𝑡−𝑧𝜕𝑢𝜕𝑧+𝑡²𝑢=0 (1.2)(1.3) 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:1−𝑡²𝜕²𝑢𝜕𝑡²+𝑧²𝜕²𝑢𝜕𝑧²−𝛼−𝛽+𝛼+𝛽+2𝑡𝜕𝑢+𝑧𝜕𝑡𝛼+𝛽+2𝜕𝑢𝜕𝑧=0, (1.4) where 𝛼,𝛽∈ℂ. Equation (1.4) has a lot of entire solutions on ℂ² 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)𝑡𝜕²𝑢𝜕𝑡²+(𝛼+1−𝑡)𝜕𝑢𝜕𝑡+𝑧𝜕𝑢𝜕𝑧=0 (1.5) for a real 𝛼>0. 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 by𝐿_𝑛(𝛼,𝑡)=^𝑛_𝑘=0𝑛+𝛼()𝑛−𝑘−𝑡^𝑘𝑘!, (1.6) which are solutions of the following ordinary differential equations (ODE):𝑡𝑑²𝜔𝑑𝑡²+(𝛼+1−𝑡)𝑑𝜔𝑑𝑡+𝑛𝜔=0. (1.7) Moreover, Hu [9] pointed out that the generating function of 𝐿_𝑛(𝛼,𝑡)𝐹(𝛼,𝑡,𝑧)=(1−𝑧)^−𝛼−1𝑒^{−𝑡𝑧/(1−𝑧)}=^∞_𝑛=0𝐿_𝑛(𝛼,𝑡)𝑧^𝑛 (1.8) 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 𝑢=𝑓(𝑡,𝑧) on ℂ², if and only if 𝑢=𝑓(𝑡,𝑧) has a series expansion 𝑓(𝑡,𝑧)=^∞_𝑛=0𝑐_𝑛𝐿_𝑛(𝛼,𝑡)𝑧^𝑛 (1.9) such that limsup_𝑛→∞^𝑛𝑐_𝑛=0. (1.10)

If 𝑓(𝑡,𝑧) is an entire function on ℂ², set 𝑀𝑟,𝑓=max_{|𝑡||𝑧|≤𝑟,≤𝑟}𝑓(𝑡,𝑧), (1.11) we define its order by ord𝑓=limsup_𝑟→∞log⁺log⁺𝑀𝑟,𝑓,log𝑟 (1.12) wherelog⁺𝑥=log𝑥,if𝑥≥1,0,if𝑥<1. (1.13)

Theorem 1.2. If 𝑓(𝑡,𝑧) is defined by (1.9) and (1.10), then 𝜌≤ord𝑓≤max1,𝜌, (1.14) where 𝜌=limsup_𝑛→∞log𝑛log1/^𝑛𝑐_𝑛. (1.15)

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 𝜌(𝜌≥1).

If 0<𝜆=ord(𝑓)<∞, we define the type of 𝑓 bytyp𝑓=limsup_𝑟→∞log⁺𝑀𝑟,𝑓𝑟^𝜆. (1.16)

Theorem 1.3. If 𝑓(𝑡,𝑧) is defined by (1.9) and (1.10), and 1<𝜆=ord(𝑓)<∞, then the type 𝜎=typ(𝑓) satisfies 𝑒𝜎𝜆=limsup_𝑛→∞𝑛^𝑛𝑐_𝑛^𝜆. (1.17)

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 𝐿_𝑛(𝛼,𝑡), which will play an important role in this paper. The following asymptotic properties of 𝐿_𝑛(𝛼,𝑡) can be found in [13]:

(a) 𝐿_𝑛()=1𝛼,𝑡2√𝜋𝑒^𝑡/2(−𝑡)^{−𝛼/2−1/4}𝑛^𝛼/2−1/4𝑒^{2√−𝑛𝑡}1+𝑂𝑛^−1/2(𝑛⟶∞) (2.1) holds for 𝑡 in the complex plane cut along the positive real semiaxis; thus, for |𝑡|≤𝑟, we obtain that 𝐿_𝑛(𝛼,𝑡)≤𝑛^𝛼/2−1/4𝑒^𝑟/2𝑟^{−𝛼/2−1/4}𝑒^2√𝑛𝑟 (2.2) holds when 𝑛 is large enough.

(b) 𝐿_𝑛(𝛼,𝑡)𝑛^𝛼/2=𝑒^𝑡/2𝑡^−𝛼/2𝐽_𝛼2√𝑛𝑡+𝑂𝑛^−3/4(𝑛⟶∞) (2.3) holds uniformly on compact subsets of (0,+∞), where 𝐽_𝛼 is the Bessel function and 𝐽_𝛼2√=2𝑛𝑡√()√𝜋Γ𝛼+1/2𝑛𝑡^𝛼₀^𝜋/2cos√𝑛𝑡cos𝑥sin^2𝛼𝑥d𝑥, (2.4) combining with (2.3), for |𝑡|≤𝑟 we can deduce that 𝐿_𝑛(𝛼,𝑡)≤𝜋𝑛^𝛼√()𝑒𝜋Γ𝛼+1/2^𝑡/2≤𝜋𝑛^𝛼√()𝑒𝜋Γ𝛼+1/2^𝑟/2≤𝑛^𝛼/2−1/4𝑒^𝑟/2𝑟^{−𝛼/2−1/4}𝑒^2√𝑛𝑟 (2.5) holds when 𝑛 is large enough. Then (2.2) and (2.5) imply 𝑀𝑟,𝐿_𝑛≤𝑛^𝛼/2−1/4𝑒^𝑟/2𝑟^{−𝛼/2−1/4}𝑒^2√𝑛𝑟, (2.6) where 𝑀𝑟,𝐿_𝑛=max_{|𝑡|≤𝑟}𝐿_𝑛(𝛼,𝑡). (2.7)

3. Proof of Theorem 1.1

Assuming that 𝑢=𝑓(𝑡,𝑧) is an entire solution on ℂ² satisfying (1.5), we have Taylor expansion𝑓(𝑡,𝑧)=^∞_𝑛=0𝑤_𝑛(𝑡)𝑧𝑛!^𝑛, (3.1) where𝑤_𝑛(𝑡)=𝜕^𝑛𝑓𝜕𝑧^𝑛(𝑡,0). (3.2) Hence 𝑤_𝑛(𝑡) is an entire solution of (1.7).

By the method of Frobenius (see [14]), we can get a second independent solution 𝑋_𝑛(𝛼,𝑡) of (1.7) which is𝑋_𝑛(𝛼,𝑡)=𝑞𝐿_𝑛(𝛼,𝑡)log𝑡+^∞_𝑖=0𝑝_𝑖𝑡^𝑖, (3.3) where 𝑞(≠0),𝑝_𝑖 are constants.

So there exist 𝑐_𝑛 and 𝑏_𝑛 satisfying𝑤_𝑛(𝑡)=𝑛!𝑐_𝑛𝐿_𝑛(𝛼,𝑡)+𝑏_𝑛𝑋_𝑛(𝛼,𝑡). (3.4) Because of the singularity of 𝑋_𝑛(𝛼,𝑡) at 𝑡=0, we obtain 𝑏_𝑛=0. That shows𝑓(𝑡,𝑧)=^∞_𝑛=0𝑐_𝑛𝐿_𝑛(𝛼,𝑡)𝑧^𝑛. (3.5)

Now we need to estimate the terms of 𝑐_𝑛. Since𝑓(0,𝑧)=^∞_𝑛=0𝑐_𝑛𝐿_𝑛(𝛼,0)𝑧^𝑛 (3.6) is an entire function, we havelimsup_𝑛→∞^𝑛𝑐_𝑛𝐿_𝑛(𝛼,0)=0. (3.7) Since𝐿_𝑛()=𝑛≈𝑛𝛼,0𝑛+𝛼^𝛼Γ(𝛼+1), (3.8) we easily getlimsup_𝑛→∞^𝑛𝑐_𝑛=0. (3.9)

Conversely, the relations (1.7), (1.9), and (1.10) imply that𝑡𝜕²𝑓𝜕𝑡²+(𝛼+1−𝑡)𝜕𝑓𝜕𝑡+𝑧𝜕𝑓=𝜕𝑧^∞_𝑛=0𝑐_𝑛𝑡𝑑²𝐿_𝑛(𝛼,𝑡)𝑑𝑡²+(𝛼+1−𝑡)𝑑𝐿_𝑛(𝛼,𝑡)𝑑𝑡+𝑛𝐿_𝑛(𝛼,𝑡)𝑧^𝑛=0 (3.10) holds for all (𝑡,𝑧)∈ℂ². Since (2.6) implieslimsup_𝑛→∞^𝑛𝑀𝑟,𝐿_𝑛≤1, (3.11) we havelimsup_𝑛→∞^𝑛𝑐_𝑛𝐿_𝑛(𝛼,𝑡)≤limsup_𝑛→∞^𝑛𝑐_𝑛𝑀𝑟,𝐿_𝑛≤limsup_𝑛→∞^𝑛𝑐_𝑛. (3.12) Combining (1.10), (3.10) with (3.12), we can get that 𝑢=𝑓(𝑡,𝑧) is obviously an entire solution of (1.5) on ℂ².

4. Proof of Theorem 1.2

Firstly, we prove 𝜌≤ord(𝑓). If 𝜌=0, the result is trivial. Now we assume 0<𝜌≤∞ and prove ord(𝑓)≥𝑘₁ for any 0<𝑘₁<𝜌. The relation (1.15) implies that there exists a sequence 𝑛_𝑗→∞ such that𝑛_𝑗log𝑛_𝑗≥𝑘₁1log𝑐_𝑛𝑗. (4.1) By using Cauchy’s inequality of holomorphic functions, we have𝜕^𝑛𝑓𝜕𝑧^𝑛(0,0)≤𝑛!𝑟^−𝑛𝑀𝑟,𝑓, (4.2) together with the formula of the coefficients of the Taylor expansion𝜕^𝑛𝑓𝜕𝑧^𝑛(0,0)=𝑐_𝑛𝐿_𝑛(𝛼,0)𝑛!, (4.3) we obtain 𝑀(𝑟,𝑓)≥|𝑐_𝑛𝐿_𝑛(𝛼,0)|𝑟^𝑛. Since |𝐿_𝑛(𝛼,0)|≥1, we have𝑐_𝑛𝑟^𝑛≤𝑀𝑟,𝑓, (4.4) thenlog𝑀𝑟,𝑓≥log𝑐_𝑛+𝑛log𝑟≥𝑛_𝑗1log𝑟−𝑘₁log𝑛_𝑗. (4.5) Putting 𝑟_𝑗=(𝑒𝑛_𝑗)^1/𝑘1, we havelog𝑀𝑟_𝑗≥𝑟,𝑓^𝑘1𝑗𝑒𝑘₁, (4.6) which means ord(𝑓)≥𝑘₁. Then we can get ord(𝑓)≥𝜌.

Next, we will prove ord(𝑓)≤max(1,𝜌). Set 𝜌^′=max(1,𝜌). The result is easy for 𝜌^′=∞; then we assume 𝜌^′<∞. For any 𝜀>0, (1.15) implies that there exists 𝑛₀>0, when 𝑛>𝑛₀, we have𝑐_𝑛<𝑛^{−𝑛(1+2𝜀)/𝜌′(1+3𝜀)}=𝑛^{−𝑛(1+2𝜀)/𝑘2}, (4.7) where 𝑘₂=𝜌^′(1+3𝜀)>1. For any 𝛼≥0, there exists 𝑛₁>𝑛₀ such that when 𝑛>𝑛₁, 𝑛^{(𝛼/2−1/4)}<𝑛^{𝑛(𝜀/𝑘2)}, (4.8) combining with (2.6) and (4.7), we get 𝑀𝑟,𝑓≤^∞_𝑛=0𝑐_𝑛𝑀𝑟,𝐿_𝑛𝑟^𝑛≤𝐶𝑟^2𝑛1+𝐶^∞_𝑛>𝑛1𝑐_𝑛𝑛^𝛼/2−1/4𝑒^𝑟/2𝑒^2√𝑛𝑟𝑟^𝑛≤𝐶𝑟^2𝑛1+𝐶𝑒^∞𝑟/2_𝑛>𝑛1𝑛^{−(1+𝜀)𝑛/𝑘2}𝑒^2√𝑛𝑟𝑟^𝑛, (4.9) where 𝐶 is a constant but not necessary to be the same every time.

Set 𝑚₁(𝑟)=((1/𝜀)(2√𝑟/log𝑟))², which means that 𝑒^2√𝑛𝑟<𝑟^𝜀𝑛 for 𝑛>𝑚₁(𝑟). Further set 𝑚₂(𝑟)=(2𝑟)^𝑘2, which yields that when 𝑛≥𝑚₂(𝑟), 𝑛^−1/𝑘2𝑟^(1+𝜀)𝑛≤12^(1+𝜀)𝑛<12^𝑛. (4.10) Obviously, we can choose 𝑟₀>0 such that 𝑚₂(𝑟)>𝑚₁(𝑟) for 𝑟>𝑟₀. Then_{𝑛>𝑚2(𝑟)}𝑛^{−(1+𝜀)𝑛/𝑘2}𝑒^2√𝑛𝑟𝑟^𝑛≤^∞_{𝑛>𝑚2(𝑟)}𝑛^{−(1+𝜀)𝑛/𝑘2}𝑟^(1+𝜀)𝑛≤^∞_{𝑛>𝑚2(𝑟)}12^𝑛≤1. (4.11) We also have_{𝑛1<𝑛≤𝑚2(𝑟)}𝑛^{−(1+𝜀)𝑛/𝑘2}𝑒^2√𝑛𝑟𝑟^𝑛≤_{𝑛1<𝑛≤𝑚2(𝑟)}𝑛^{−(1+𝜀)𝑛/𝑘2}𝑒^{2𝑚2(𝑟)𝑟}𝑟^𝑚2(𝑟)≤_{𝑛1<𝑛≤𝑚2(𝑟)}𝑛^{−(1+𝜀)𝑛/𝑘2}𝑟^𝑚2(𝑟)𝑟^{𝜀𝑚2(𝑟)}≤𝑟^{(1+𝜀)𝑚2(𝑟)}_{𝑛1<𝑛≤𝑚2(𝑟)}𝑛^{−(1+𝜀)𝑛/𝑘2}≤𝐶𝑟^{(1+𝜀)𝑚2(𝑟)}=𝐶𝑟^{(1+𝜀)(2𝑟)𝑘2}. (4.12) Therefore, when |𝑡|=𝑟≥𝑟₀, we have𝑀𝑟,𝑓≤𝐶𝑟^2𝑛1+𝐶𝑒^𝑟/2+𝐶𝑒^𝑟/2𝑟^{(1+𝜀)(2𝑟)𝑘2}, (4.13) which means ord(𝑓)≤𝑘₂. Hence ord(𝑓)≤𝜌^′=max(1,𝜌) follows by letting 𝜀→0.

5. Proof of Theorem 1.3

Set𝜅=limsup_𝑛→∞𝑛^𝑛𝑐_𝑛^𝜆. (5.1) At first, we prove 𝑒𝜆𝜎≥𝜅. The result is trivial for 𝜅=0, we assume 0<𝜅≤∞ and take 𝜀 with 0<𝜀<𝜅, set𝑘₃=1𝜅−𝜀,if𝜅<∞,𝜀,if𝜅=∞. (5.2) Equation (5.1) implies that there exists a sequence 𝑛_𝑗→∞ satisfying𝑐_𝑛𝑗>𝑘₃𝑛_𝑗^{𝑛𝑗/𝜆}, (5.3) combining with (4.4), we can deduce that𝑀𝑟,𝑓≥𝑐_𝑛𝑗𝑟^𝑛𝑗≥𝑘₃𝑛_𝑗^{𝑛𝑗/𝜆}𝑟^𝑛𝑗=𝑘₃𝑛_𝑗𝑟^{𝜆𝑛𝑗/𝜆}. (5.4) Taking 𝑟^𝜆_𝑗=𝑒𝑛_𝑗/𝑘₃, we get 𝑀(𝑟_𝑗,𝑓)>𝑒^{𝑘3𝑟𝜆𝑗/𝑒𝜆}, which yields 𝜎≥𝑘₃/𝑒𝜆, so 𝑒𝜆𝜎≥𝜅.

Next, we prove 𝑒𝜆𝜎≤𝜅. We may assume 𝜅<∞. Equation (5.1) implies that for any 𝜀>0, there exists 𝑛₀>0, such that when 𝑛>𝑛₀,𝑐_𝑛<𝜅+(𝜀/2)𝑛^𝑛/𝜆. (5.5) For any 𝛼≥0, we choose 𝑛₁(>𝑛₀) such that when 𝑛>𝑛₁,𝑛^𝛼/2−1/4<𝜅+𝜀𝜅+(𝜀/2)^𝑛/𝜆, (5.6) combining with (2.6), we have𝑀𝑟,𝑓≤^∞_𝑛=0𝑐_𝑛𝑀𝑟,𝐿_𝑛𝑟^𝑛≤𝐶𝑟^2𝑛1+𝐶^∞_𝑛>𝑛1𝑐_𝑛𝑛^𝛼/2−1/4𝑒^𝑟/2𝑒^2√𝑛𝑟𝑟^𝑛≤𝐶𝑟^2𝑛1+𝐶𝑒^∞𝑟/2_𝑛>𝑛1𝜅+𝜀𝑛^𝑛/𝜆𝑒^2√𝑛𝑟𝑟^𝑛. (5.7)

Set 𝑚₃(𝑟)=16𝑟𝜆², when 𝑛>𝑚₃(𝑟), we deduce 𝑒^2√𝑛𝑟<𝑒^𝑛/2𝜆. Set 𝑚₄(𝑟)=2(𝜅+𝜀)𝑟^𝜆, it is obvious that (𝜅+𝜀)𝑟^𝜆/𝑛<1/2 for 𝑛>𝑚₄(𝑟). Since 𝜆>1, there exists 𝑟₁ such that when 𝑟>𝑟₁,𝑚₄(𝑟)=2(𝜅+𝜀)𝑟^𝜆>16𝑟𝜆²=𝑚₃(𝑟). (5.8) Then^∞_{𝑛>𝑚4(𝑟)}𝜅+𝜀𝑛^𝑛/𝜆𝑒^2√𝑛𝑟𝑟^𝑛=^∞_{𝑛>𝑚4(𝑟)}(𝜅+𝜀)𝑟^𝜆𝑛^𝑛/𝜆𝑒^2√𝑛𝑟≤^∞_{𝑛>𝑚4(𝑟)}𝑒4^𝑛/2𝜆≤𝐶. (5.9) We note that for 𝑎>0, 𝑏>0, max_𝑥>0(𝑎/𝑥)^𝑥/𝑏=𝑒^{𝑎/𝑒𝑏}, then we have𝜅+𝜀𝑛^𝑛/𝜆𝑟^𝑛=(𝜅+𝜀)𝑟^𝜆𝑛^𝑛/𝜆≤𝑒^{(𝜅+𝜀)𝑟𝜆/𝑒𝜆}. (5.10) This shows_{𝑛1<𝑛≤𝑚4(𝑟)}𝜅+𝜀𝑛^𝑛/𝜆𝑒^2√𝑛𝑟𝑟^𝑛≤𝑚₄(𝑟)𝑒^{2𝑚4(𝑟)𝑟}𝑒^{(𝜅+𝜀)𝑟𝜆/𝑒𝜆}≤2(𝜅+𝜀)𝑟^𝜆𝑒^{22(𝜅+𝜀)𝑟𝜆+1}𝑒^{(𝜅+𝜀)𝑟𝜆/𝑒𝜆}. (5.11)

Therefore when |𝑡|=𝑟≥𝑟₁,𝑀𝑟,𝑓≤𝐶𝑒^𝑟/2(𝜅+𝜀)𝑟^𝜆𝑒^{22(𝜅+𝜀)𝑟𝜆+1}𝑒^{(𝜅+𝜀)𝑟𝜆/𝑒𝜆}+𝐶𝑟^2𝑛1+𝐶𝑒^𝑟/2. (5.12) Together with 𝜆>1 and the definition of type, we can get 𝜎≤(𝜅+𝜀)/𝑒𝜆, which yields 𝑒𝜆𝜎≤𝜅 by letting 𝜀→0.

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.