Approximation of Analytic Functions by Kummer Functions

Abstract

We solve the inhomogeneous Kummer differential equation of the form and apply this result to the proof of a local Hyers-Ulam stability of the Kummer differential equation in a special class of analytic functions.

1. Introduction

Assume that and are a topological vector space and a normed space, respectively, and that is an open subset of . If for any function satisfying the differential inequality

(1.1)

for all and for some , there exists a solution of the differential equation

(1.2)

such that for any , where depends on only, then we say that the above differential equation satisfies the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain is not the whole space ). We may apply this terminology for other differential equations. For more detailed definition of the Hyers-Ulam stability, refer to [1–6].

Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [7, 8]). Here, we will introduce a result of Alsina and Ger (see [9]). If a differentiable function is a solution of the differential inequality , where is an open subinterval of , then there exists a solution of the differential equation such that for any .

This result of Alsina and Ger has been generalized by Takahasi et al.. They proved in [10] that the Hyers-Ulam stability holds true for the Banach space valued differential equation (see also [11]).

Using the conventional power series method, the author [12] investigated the general solution of the inhomogeneous Legendre differential equation of the form

(1.3)

under some specific conditions, where is a real number and the convergence radius of the power series is positive. Moreover, he applied this result to prove that every analytic function can be approximated in a neighborhood of by the Legendre function with an error bound expressed by (see [13–16]).

In Section 2 of this paper, employing power series method, we will determine the general solution of the inhomogeneous Kummer (differential) equation

(1.4)

where and are constants and the coefficients of the power series are given such that the radius of convergence is , whose value is in general permitted to be infinite. Moreover, using the idea from [12, 13, 15], we will prove the Hyers-Ulam stability of the Kummer's equation in a class of special analytic functions (see the class in Section 3).

In this paper, and denote the set of all nonnegative integers and the set of all integers, respectively. For each real number , we use the notation to denote the ceiling of , that is, the least integer not less than .

2. General Solution of (1.4)

The Kummer (differential) equation

(2.1)

which is also called the confluent hypergeometric differential equation, appears frequently in practical problems and applications. The Kummer's equation (2.1) has a regular singularity at and an irregular singularity at . A power series solution of (2.1) is given by

(2.2)

where is the factorial function defined by and for all . The above power series solution is called the Kummer function or the confluent hypergeometric function. We know that if neither nor is a nonpositive integer, then the power series for converges for all values of .

Let us define

(2.3)

We know that if then and are independent solutions of the Kummer's equation (2.1). When is not defined at because of the factor in the above definition of .

By considering this fact, we define

(2.4)

for any . It should be remarked that if and both and are not nonpositive integers, then and converge for all (see [17, Section ]).

Theorem 2.1.

Let and be real constants such that and neither nor is a nonpositive integer. Assume that the radius of convergence of the power series is and that there exists a real number with

(2.5)

for all sufficiently large integers . Let us define and . Then, every solution of the inhomogeneous Kummer's equation (1.4) can be represented by

(2.6)

where is a solution of the Kummer's equation (2.1).

Proof.

Assume that a function is given by (2.6). We first prove that the function , defined by , satisfies the inhomogeneous Kummer's equation (1.4). Since

(2.7)

we have

(2.8)

which proves that is a particular solution of the inhomogeneous Kummer's equation (1.4).

We now apply the ratio test to the power series expression of as follows:

(2.9)

Then, it follows from (2.5) that

(2.10)

Therefore, the power series expression of converges for all . Moreover, the convergence region of the power series for is the same as those of power series for and . In this paper, the convergence region will denote the maximum open set where the relevant power series converges. Hence, the power series expression for has the same convergence region as that of . This implies that is well defined on and so does for in (2.6) because converges for all under our hypotheses for and (see above Theorem 2.1).

Since every solution to (1.4) can be expressed as a sum of a solution of the homogeneous equation and a particular solution of the inhomogeneous equation, every solution of (1.4) is certainly in the form of (2.6).

Remark 2.2.

We fix and and we define

(2.11)

for every . Then, since , there exists a real number such that

(2.12)

for all sufficiently large integers . Hence, the sequence satisfies condition (2.5) for all sufficiently large integers .

3. Hyers-Ulam Stability of (2.1)

In this section, let and be real constants and assume that is a constant with . For a given , let us denote the set of all functions with the properties (a) and (b):

(a) is represented by a power series whose radius of convergence is at least ;

(b)it holds true that for all , where for each .

It should be remarked that the power series in (b) has the same radius of convergence as that of given in (a).

In the following theorem, we will prove a local Hyers-Ulam stability of the Kummer's equation under some additional conditions. More precisely, if an analytic function satisfies some conditions given in the following theorem, then it can be approximated by a "combination" of Kummer functions such as and (see the first part of Section 2).

Theorem 3.1.

Let and be real constants such that and neither nor is a nonpositive integer. Suppose a function is representable by a power series whose radius of convergence is at least . Assume that there exist nonnegative constants and satisfying the condition

(3.1)

for all , where . Indeed, it is sufficient for the first inequality in (3.1) to hold true for all sufficiently large integers . Let us define . If and it satisfies the differential inequality

(3.2)

for all and for some , then there exists a solution of the Kummer's equation (2.1) such that

(3.3)

for any , where .

Proof.

By the definition of , we have

(3.4)

for all . So by (3.2) we have

(3.5)

for any . Since , this inequality together with (b) yields

(3.6)

for each .

By Abel's formula (see [18, Theorem ]), we have

(3.7)

for any and . With ( is the ceiling of ), we know that

(3.8)

Due to (3.4), it follows from Theorem 2.1 and (2.6) that there exists a solution of the Kummer's equation (2.1) such that

(3.9)

for all . By using (3.1), (3.6), (3.7), and (3.8), we can estimate

(3.10)

for all .

We now assume a stronger condition, in comparison with (3.1), to approximate the given function by a solution of the Kummer's equation on a larger (punctured) interval.

Corollary 3.2.

Let and be real constants such that and neither nor is a nonpositive integer. Suppose a function is representable by a power series which converges for all . For every , let us define . Moreover, assume that

(3.11)

and there exists a nonnegative constant satisfying

(3.12)

for all . If and it satisfies the differential inequality (3.2) for all and for some , then there exists a solution of the Kummer's equation (2.1) such that

(3.13)

for any , where and is a sufficiently large integer.

Proof.

In view of (3.11) and (3.12), we can choose a sufficiently large integer with

(3.14)

where the first inequality holds true for all sufficiently large and the second one holds true for all .

If we define , then Theorem 3.1 implies that there exists a solution of the Kummer's equation such that the inequality given for holds true for any .

4. An Example

We fix , , and . And we define

(4.1)

for all , where we set . We further define

(4.2)

for any .

Then, we set , that is,

(4.3)

for every . Obviously, all s are positive, and the sequence is strictly monotone decreasing, from the 4th term on, to . More precisely, .

Since

(4.4)

we get

(4.5)

for each and

(4.6)

for all . Hence, we obtain

(4.7)

for any , where , implying that .

We will now show that satisfies condition (3.1). For any , we have

(4.8)

since .

It follows from (4.8) that

(4.9)

We know that the inequality (4.9) is also true for .

On the other hand, in view of Remark 2.2, there exists a constant such that inequality (2.12) holds true for all sufficiently large integers . By (2.12) and (4.9), we conclude that satisfies condition (3.1) with .

Finally, it follows from (4.6) that

(4.10)

for all with .

According to Theorem 3.1, there exists a solution of the Kummer's equation (2.1) such that

(4.11)

for all .