Error Bounds for Asymptotic Solutions of Second-Order Linear Difference Equations II: The First Case

Abstract

We discuss in detail the error bounds for asymptotic solutions of second-order linear difference equation where and are integers, and have asymptotic expansions of the form , , for large values of , , and .

1. Introduction

Asymptotic expansion of solutions to second-order linear difference equations is an old subject. The earliest work as we know can go back to 1911 when Birkhoff [1] first deal with this problem. More than eighty years later, this problem was picked up again by Wong and Li [2, 3]. This time two papers on asymptotic solutions to the following difference equations:

(1.1)

(1.2)

were published, respectively, where coefficients and have asymptotic properties

(1.3)

for large values of , , , and .

Unlike the method used by Olver [4] to treat asymptotic solutions of second-order linear differential equations, the method used in Wong and Li's papers cannot give us way to obtain error bounds of these asymptotic solutions. Only order estimations were given in their papers. The estimations of error bounds for these asymptotic solutions to (1.1) were given in [5] by Zhang et al. But the problem of obtaining error bounds for these asymptotic solutions to (1.2) is still open. The purpose of this and the next paper (Error bounds for asymptotic solutions of second-order linear difference equations II: the second case) is to estimate error bounds for solutions to (1.2). The idea used in this paper is similar to that of Olver to obtain error bounds to the Liouville-Green (WKB) asymptotic expansion of solutions to second-order differential equations. It should be pointed out that similar method appeared in some early papers, such as Spigler and Vianello's papers [6–9].

In Wong and Li's second paper [3], two different cases were given according to different values of parameters. The first case is devoted to the situation when , and in the second case as where . The whole proof of the result is too long to understand, so we divide the estimations into two parts, part I (this paper) and part II (the next paper), which correspond to the different two cases of [3], respectively.

In the rest of this section, we introduce the main results of [3] in the case that is positive. In the next section, we give two lemmas on estimations of bounds for solutions to a special summation equation and a first order nonlinear difference equation which will be often used later. Section 3 is devoted to the case when . And in Section 4, we discuss the case when . The next paper (Error bounds for asymptotic solutions of second-order linear difference equations II: the second case) is dedicated to the case when .

1.1. The Result in [3] When  

When , from [3] we know that (1.2) has two linearly independent solution and

(1.4)

(1.5)

(1.6)

(1.7)

for .

1.2. The Result in [3] When  

When , from [3] we know that (1.2) has two linearly independent solutions and

(1.8)

(1.9)

(1.10)

(1.11)

In the following sections, we will discuss in detail the error bounds of the proceeding asymptotic solutions of (1.2). Before discussing the error bounds, we consider some lemmas.

2. Lemmas

2.1. The Bounds for Solutions to the Summation Equation

We consider firstly a bound of a special solution for the "summary equation"

(2.1)

Lemma 2.1.

Let , , , be real or complex functions of integer variables ; and are integers. If there exist nonnegative constants , , , , , , β, , , , , , which satisfy

(2.2)

and when ,

(2.3)

where and are positive functions of integer variable . Let , be integers defined by

(2.4)

then (2.1) has a solution , which satisfies

(2.5)

for .

Proof.

Set

(2.6)

then

(2.7)

The inequality , is used here. Assuming that

(2.8)

where

(2.9)

then

(2.10)

By induction, the inequality holds for any integer . Hence the series

(2.11)

when , that is, , is uniformly convergent in where

(2.12)

And its sum

(2.13)

satisfies

(2.14)

So we get the bound of any solution for the "summary equation" (2.1). Next we consider a nonlinear first-order difference equation.

2.2. The Bound Estimate of a Solution to a Nonlinear First-Order Difference Equation

Lemma 2.2.

If the function satisfies

(2.15)

where ( and are constants), when is large enough, then the following first-order difference equation

(2.16)

has a solution such that is bounded by a constant , when is big enough.

Proof.

Obviously from the conditions of this lemma, we know that infinite products and are convergent.

(2.17)

is a solution of (2.16) with the infinite condition. Let ; then when is large enough,

(2.18)

3. Error Bounds in the Case When  

Before giving the estimations of error bounds of solutions to (1.2), we rewrite as

(3.1)

with

(3.2)

and , , being error terms. Then , , satisfy inhomogeneous second-order linear difference equations

(3.3)

where

(3.4)

We know from [3] that

(3.5)

3.1. The Error Bound for the Asymptotic Expansion of  

Now we firstly estimate the error bound of the asymptotic expansion of in the case . Let

(3.6)

It can be easily verified that

(3.7)

are two linear independent solutions of the comparative difference equation

(3.8)

From the definition, we know that the two-term approximation of is

(3.9)

where is the reminder and the coefficient of is zero. So is a constant. And satisfies being a constant; here we have made use of the definitions of in (1.5), (1.7), and .

Equation (3.8) is a second-order linear difference equation with two known linear independent solutions. Its coefficients are quite similar to those in (3.3). This reminds us to rewrite (3.3) in the form similar to (3.8).

According to the coefficients in (3.8), we rewrite (3.3) as

(3.10)

where and are such that

(3.11)

are finite. Equation (3.10) is a inhomogeneous second-order linear difference equation; its solution takes the form of a particular solution added to an arbitrary linear combination of solutions to the associated homogeneous linear difference equation(3.8).

From [10], any solution of the "summary equation"

(3.12)

is a solution of (3.10), where

(3.13)

Now we estimate the bound of the function .

Firstly we consider the denominator in . We get from(3.8)

(3.14)

Set the Wronskian of the two solutions of the comparative difference equation as

(3.15)

we have

(3.16)

From (3.16), we have

(3.17)

From Lemma 3 of [5], we obtain

(3.18)

where

(3.19)

(3.20)

is an integer which is large enough such that , when .

Let , for the property of , we know that is a constant. Then we obtain from (3.18)

(3.21)

Now considering the numerator in , we get

(3.22)

Here we have made use of .

From Lemma  2 of [5], we have

(3.23)

where is a constant. For the bound of , we set

(3.24)

then

(3.25)

where

(3.26)

By simple calculations, we get

(3.27)

Here we have made use of (1.5) and (1.7).

Since , we have

(3.28)

Here we also have made use of (1.5) and (1.7).

Let

(3.29)

we have from (3.24) the bound of

(3.30)

For the bound of , set , , , , , , , , , , ; we have from Lemma 2.1 that

(3.31)

when

(3.32)

that is, and .

3.2. The Error Bound for the Asymptotic Expansion of ()

Now we estimate the error bound of the asymptotic expansion of the linear independent solution to the original difference equation as . Let

(3.33)

From (3.3), we have

(3.34)

For being a solution of (1.2), let

(3.35)

then satisfies the first-order linear difference equation

(3.36)

The solution of (3.36) is

(3.37)

where is a constant, and is an integer which is large enough such that when ,

(3.38)

The two-term approximation of is

(3.39)

where is the reminder and is a constant.

From Lemma 3 of [5], we obtain

(3.40)

where

(3.41)

are constants.

Substituting (3.38) and (3.40) into (3.37), we get

(3.42)

Let then

(3.43)

From (3.35), we have

(3.44)

where is a constant. Let ; we have

(3.45)

For , there exists a positive integer such that

(3.46)

when . Thus the sequence is increasing when .

Let ; then

(3.47)

where Hence

(3.48)

From (3.33), we obtain

(3.49)

Thus we complete the estimate of error bounds to asymptotic expansions of solutions of (1.2) as .

4. Error Bounds in Case When  

Here we also rewrite as

(4.1)

with

(4.2)

and , ,2, are error terms. Then , ,2, satisfy the inhomogeneous second-order linear difference equations

(4.3)

where

(4.4)

We know from [3] that

(4.5)

4.1. The Error Bound for the Asymptotic Expansion of  

Now let us come to the case when . This time a difference equation which has two known linear independent solutions is also constructed for the purpose of comparison for (1.2).

Since

(4.6)

where

(4.7)

is a constant and ( is a constant), from Lemma 2.2, we know the difference equation

(4.8)

with condition having a solution such that

(4.9)

is a constant. And the function

(4.10)

such that

(4.11)

is a constant. Here we have made use of the definitions of , , , in (1.9), (1.11) and .

Obviously functions

(4.12)

are two linear independent solutions of the difference equation

(4.13)

This difference equation(4.13) can be regarded as the comparative equation of (4.3). Rewriting (4.3) in the form similar to the comparative difference equation (4.13), we get

(4.14)

where has the property that is a constant. Equation (4.14) is an inhomogeneous second-order linear difference equation; its solution takes the form of a particular solution added to an arbitrary linear combination of solutions to the associated homogeneous linear difference equation (4.13).

From [10], any solution of the "summary equation"

(4.15)

where

(4.16)

is a solution of (4.14).

Similar to Section 3.1, we have

(4.17)

Let

(4.18)

we get

(4.19)

Set , , , , , , , , , , , ; we have from Lemma 2.1 that

(4.20)

when

(4.21)

that is,

,  .

4.2. The Error Bound for the Asymptotic Expansion of  

Let

(4.22)

From (3.3), we have

(4.23)

Using the method employed in Section 3.2, it is not difficult to obtain

(4.24)

Now we completed the estimate of the error bounds for asymptotic solutions to second order linear difference equations in the first case. For the second case, we leave it to the second part of this paper: Error Bound for Asymptotic Solutions of Second-order Linear Difference Equation II: the second case.

In the rest of this paper, we would like to give an example to show how to use the results of this paper to obtain error bounds of asymptotic solutons to second-order linear difference equations. Here the difference equation is

(4.25)

It is a special case of the equation

(4.26)

, which is satisfied by Tricomi-Carlitz polynomials. By calculation, the constant in (3.30) is . So (4.25) has a solution

(4.27)

for with the error term satisfing

(4.28)