Approximation of Analytic Functions by Bessel's Functions of Fractional Order

Abstract

We will solve the inhomogeneous Bessel's differential equation 𝑥²𝑦′′(𝑥)+𝑥𝑦^′(𝑥)+(𝑥²−𝜈²)𝑦(𝑥)=∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚, where 𝜈 is a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.

1. Introduction

The stability problem for functional equations starts from the famous talk of Ulam and the partial solution of Hyers to the Ulam problem (see [1, 2]). Thereafter, Rassias [3] attempted to solve the stability problem of the Cauchy additive functional equation in a more general setting.

The stability concept introduced by Rassias's theorem significantly influenced a number of mathematicians to investigate the stability problems for various functional equations (see [4–14] and the references therein).

Assume that 𝑌 is a normed space and 𝐼 is an open subset of ℝ. If for any function 𝑓∶𝐼→𝑌 satisfying the differential inequality𝑎_𝑛(𝑥)𝑦^(𝑛)(𝑥)+𝑎_𝑛−1(𝑥)𝑦^(𝑛−1)(𝑥)+⋯+𝑎₁(𝑥)𝑦^′(𝑥)+𝑎₀(𝑥)𝑦(𝑥)+ℎ(𝑥)≤𝜀, (1.1) for all 𝑥∈𝐼 and for some 𝜀≥0, there exists a solution 𝑓₀∶𝐼→𝑌 of the differential equation 𝑎_𝑛(𝑥)𝑦^(𝑛)(𝑥)+𝑎_𝑛−1(𝑥)𝑦^(𝑛−1)(𝑥)+⋯+𝑎₁(𝑥)𝑦^′(𝑥)+𝑎₀(𝑥)𝑦(𝑥)+ℎ(𝑥)=0, (1.2) such that ‖𝑓(𝑥)−𝑓₀(𝑥)‖≤𝐾(𝜀) for any 𝑥∈𝐼, where 𝐾(𝜀) depends on 𝜀 only, then we say that the above differential equation satisfies the Hyers-Ulam stability (or the local Hyers-Ulam stability if the domain 𝐼 is not the whole space ℝ). We may apply these terminologies for other differential equations. For more detailed definition of the Hyers-Ulam stability, refer to [2, 3, 6, 8, 10–12, 14].

Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [15, 16]). Here, we will introduce a result of Alsina and Ger (see [17]): If a differentiable function 𝑓∶𝐼→ℝ is a solution of the differential inequality |𝑦^′(𝑥)−𝑦(𝑥)|≤𝜀, where 𝐼 is an open subinterval of ℝ, then there exists a constant 𝑐 such that |𝑓(𝑥)−𝑐𝑒^𝑥|≤3𝜀 for any 𝑥∈𝐼.

This result of Alsina and Ger has been generalized by Takahasi et al. They proved in [18] that the Hyers-Ulam stability holds for the Banach space-valued differential equation 𝑦^′(𝑥)=𝜆𝑦(𝑥) (see also [19]).

Using the conventional power series method, the author has investigated the general solution of the inhomogeneous Legendre differential equation under some specific condition, and this result was applied to prove the Hyers-Ulam stability of the Legendre differential equation (see [20]). In a recent paper, he has also investigated an approximation property of analytic functions by the Legendre functions (see [21]). This study has been continued to various special functions including the Airy functions, the exponential functions, the Hermite functions, and the power functions (see [22–25]).

Recently, the author and Kim tried to prove the Hyers-Ulam stability of the Bessel differential equation𝑥²𝑦^′′(𝑥)+𝑥𝑦^′(𝑥)+𝑥²−𝜈²𝑦(𝑥)=0. (1.3) However, the obtained theorem unfortunately does not describe the Hyers-Ulam stability of the Bessel differential equation in a strict sense (see [26]).

In Section 2 of this paper, by using the ideas from [21], we will determine the general solution of the inhomogeneous Bessel differential equation𝑥²𝑦^′′(𝑥)+𝑥𝑦^′(𝑥)+𝑥²−𝜈²𝑦(𝑥)=^∞_𝑚=0𝑎_𝑚𝑥^𝑚, (1.4) where the parameter 𝜈 is a positive nonintegral number. Section 3 will be devoted to the investigation of an approximation property of the Bessel functions.

Throughout this paper, we denote by [𝑥] the largest integer not exceeding 𝑥 for any 𝑥∈ℝ, and we define 𝐼_𝜌=(−𝜌,0)∪(0,𝜌) for any 𝜌>0.

2. Inhomogeneous Bessel's Differential Equation

A function is called a Bessel function (of fractional order) if it is a solution of the Bessel differential equation (1.3), where 𝜈 is a positive nonintegral number. The Bessel differential equation plays a great role in physics and engineering. In particular, this equation is most useful for treating the boundary value problems exhibiting cylindrical symmetries.

The convergence of the power series ∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚 seems not to guarantee the existence of solutions to the inhomogeneous Bessel differential equation (1.4). Thus, we adopt an additional condition to ensure the existence of solutions to the equation.

Theorem 2.1. Let 𝜈 be a positive nonintegral number, and let 𝜌 be a positive constant. Assume that the radius of convergence of power series ∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚 is at least 𝜌 and there exists a constant 𝜎>0 satisfying the condition 𝑎_𝑚+2≤𝑚²𝜎²𝑐_𝑚, (2.1) for all sufficiently large integers 𝑚, where 𝑐_𝑚=−^[𝑚/2]_𝑖=0𝑎_2𝑖^[𝑚/2]_𝑗=𝑖1𝜈²−2𝑗²(),−foreven𝑚_{[𝑚/2]𝑖=0}𝑎_2𝑖+1^[𝑚/2]_𝑗=𝑖1𝜈²−2𝑗+1²(forodd𝑚), (2.2) for all 𝑚∈ℕ₀. Let 𝜌₀=min{𝜌,𝜎}. Then every solution 𝑦∶𝐼_𝜌0→ℂ of the Bessel's differential equation (1.4) can be expressed by 𝑦(𝑥)=𝑦_ℎ(𝑥)+^∞_𝑚=0𝑐_𝑚𝑥^𝑚, (2.3) for all 𝑥∈𝐼_𝜌0, where 𝑦_ℎ(𝑥) is a solution of the homogeneous Bessel equation (1.3).

Proof. We assume that 𝑦∶𝐼_𝜌0→ℂ is a function given in the form (2.3), and we define 𝑦_𝑝(𝑥)=𝑦(𝑥)−𝑦_ℎ(𝑥)=∑^∞_𝑚=0𝑐_𝑚𝑥^𝑚. Then, it follows from (2.1) and (2.2) that lim_𝑚→∞𝑐_𝑚+2𝑐_𝑚=lim_𝑚→∞1(𝑚+2)²−𝜈²𝑎_𝑚+2𝑐_𝑚≤1−1𝜎², (2.4) since we can deduce the relation 𝑐_𝑚+2=(𝑎_𝑚+2−𝑐_𝑚)/((𝑚+2)²−𝜈²) from (2.2) by some manipulations. That is, the power series for 𝑦_𝑝(𝑥) converges for all 𝑥∈𝐼_𝜌0. Hence, we see that the domain of 𝑦(𝑥) is well defined.
We now prove that the function 𝑦_𝑝(𝑥) satisfies the inhomogeneous equation (1.4). Indeed, it follows from (2.2) that 𝑥²𝑦_𝑝^′′(𝑥)+𝑥𝑦^′_𝑝(𝑥)+𝑥²−𝜈²𝑦_𝑝(𝑥)=^∞_𝑚=2𝑚(𝑚−1)𝑐_𝑚𝑥^𝑚+^∞_𝑚=1𝑚𝑐_𝑚𝑥^𝑚+^∞_𝑚=0𝑐_𝑚𝑥^𝑚+2−^∞_𝑚=0𝜈²𝑐_𝑚𝑥^𝑚=𝑐₁𝑥−𝜈²𝑐₀−𝜈²𝑐₁𝑥+^∞_𝑚=2𝑐_𝑚−2+𝑚²−𝜈²𝑐_𝑚𝑥^𝑚=𝑎₀+𝑎₁𝑥+^∞_𝑚=2𝑎_𝑚𝑥^𝑚, (2.5) since we obtain 𝑐₀1=−𝜈²𝑎₀,𝑐₁=11−𝜈²𝑎₁,𝑐_𝑚−2+𝑚²−𝜈²𝑐_𝑚=𝑎_𝑚,for𝑚≥2, (2.6) which proves that 𝑦_𝑝(𝑥) is a particular solution of the inhomogeneous equation (1.4).
On the other hand, since every solution to (1.4) can be expressed as a sum of a solution 𝑦_ℎ(𝑥) of the homogeneous equation and a particular solution 𝑦_𝑝(𝑥) of the inhomogeneous equation, every solution of (1.4) is certainly of the form (2.3).

3. Approximate Bessel's Differential Equation

In this section, assume that 𝜈 is a positive nonintegral number and 𝜌 is a positive constant. For a given 𝐾≥0, we denote by 𝒞_𝐾 the set of all functions 𝑦∶𝐼_𝜌→ℂ with the properties (a) and (b): (a)𝑦(𝑥) is expressible by a power series ∑^∞_𝑚=0𝑏_𝑚𝑥^𝑚 whose radius of convergence is at least 𝜌; (b)∑^∞_𝑚=0|𝑎_𝑚𝑥^𝑚|≤𝐾|∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚| for any 𝑥∈𝐼_𝜌, where 𝑎_𝑚=𝑏_𝑚−2+(𝑚²−𝜈²)𝑏_𝑚 for all 𝑚∈ℕ₀ and set 𝑏₋₂=𝑏₋₁=0.

For a positive nonintegral number 𝜈, define 𝑀_𝑒(𝑥)=max^𝑘_𝑗=𝑖𝑥²𝜈²−2𝑗²,𝑀∶0≤𝑖≤𝑘≤𝜇_𝑜(𝑥)=max^𝑘_𝑗=𝑖𝑥²𝜈²−2𝑗+1²,𝑀(𝑥)𝑀∶0≤𝑖≤𝑘≤𝜇=max_𝑒(𝑥),𝑀_𝑜(𝑥),1, (3.1) where 𝜇=[√𝜈²+𝑥²/2] and 𝐿_𝜈=^∞_𝑚=01(𝑚−𝜈)²<∞. (3.2) We remark that 𝑀(𝑥)→1 as |𝑥|→0.

We will now solve the approximate Bessel differential equations in a special class of analytic functions, 𝒞_𝐾.

Theorem 3.1. Let 𝜈 be a positive nonintegral number, and let 𝑝 be a nonnegative integer with 𝑝<𝜈<𝑝+1. Assume that a function 𝑦∈𝒞_𝐾 satisfies the differential inequality 𝑥²𝑦^′′(𝑥)+𝑥𝑦^′(𝑥)+𝑥²−𝜈²𝑦(𝑥)≤𝜀, (3.3) for all 𝑥∈𝐼_𝜌 and for some 𝜀≥0. If the sequence {𝑏_𝑚} satisfies the condition 𝑏_𝑚+2=𝑂𝑏_𝑚as𝑚⟶∞ (3.4) with a Landau constant 𝐶≥0, then there exists a solution 𝑦_ℎ(x) of the Bessel differential equation (1.3) such that 𝑦(𝑥)−𝑦_ℎ(𝑥)≤𝐾𝐿_𝜈𝑀(𝑥)𝜀, (3.5) for any 𝑥∈𝐼_𝜌0, where 𝜌₀=min{𝜌,1/√𝐶^∗} and 𝐶^∗ is a positive number larger than 𝐶. If 𝐶 and 𝜌 are sufficiently small and large; respectively, then 𝑀(𝑥)≤max|𝑥|^|𝑥|+2|𝜈²−𝑝²|^|𝑥|/2+1,|𝑥|^|𝑥|+2𝜈²−𝑝+1^2|𝑥|/2+1 (3.6) for all sufficiently large |𝑥|.

Proof. Since 𝑦 belongs to 𝒞_𝐾, it follows from (a) and (b) that 𝑥²𝑦^′′(𝑥)+𝑥𝑦^′(𝑥)+𝑥²−𝜈²𝑦(𝑥)=^∞_𝑚=0𝑏_𝑚−2+𝑚²−𝜈²𝑏_𝑚𝑥^𝑚=^∞_𝑚=0𝑎_𝑚𝑥^𝑚, (3.7) for all 𝑥∈𝐼_𝜌. By considering (3.3) and (3.7), we get ^∞_𝑚=0𝑎_𝑚𝑥^𝑚≤𝜀, (3.8) for any 𝑥∈𝐼_𝜌. This inequality, together with (b), yields ^∞_𝑚=0𝑎_𝑚𝑥^𝑚≤𝐾^∞_𝑚=0𝑎_𝑚𝑥^𝑚≤𝐾𝜀, (3.9) for each 𝑥∈𝐼_𝜌.
Now, it follows from (b) that ^𝑛_𝑖=0𝑎^𝑛_{2𝑖𝑗=𝑖}1𝜈²−2𝑗²=^𝑛_𝑖=0𝑏^𝑛_{2𝑖−2𝑗=𝑖}1𝜈²−2𝑗²−^𝑛_𝑖=0𝑏^𝑛_{2𝑖𝑗=𝑖+1}1𝜈²−2𝑗²=^𝑛−1_𝑖=−1𝑏^𝑛_{2𝑖𝑗=𝑖+1}1𝜈²−2𝑗²−^𝑛_𝑖=0𝑏^𝑛_{2𝑖𝑗=𝑖+1}1𝜈²−2𝑗²=𝑏^𝑛_−2𝑗=01𝜈²−2𝑗²−𝑏_2𝑛=−𝑏_2𝑛, (3.10) since 𝑏₋₂=0. Similarly, we obtain ^𝑛_𝑖=0𝑎^𝑛_{2𝑖+1𝑗=𝑖}1𝜈²−2𝑗+1²=−𝑏_2𝑛+1, (3.11) for all 𝑛∈ℕ₀, that is, 𝑏_𝑚=−^[𝑚/2]_𝑖=0𝑎_2𝑖^[𝑚/2]_𝑗=𝑖1𝜈²−2𝑗²(),−foreven𝑚_{[𝑚/2]𝑖=0}𝑎_2𝑖+1^[𝑚/2]_𝑗=𝑖1𝜈²−2𝑗+1²(forodd𝑚), (3.12) for all 𝑚∈ℕ₀.
On the other hand, by (b) and (3.4), we have 𝑎_𝑚+2=𝑏_𝑚+(𝑚+2)²−𝜈²𝑏_𝑚+2≤𝑏_𝑚+(𝑚+2)²𝑏_𝑚+2≤𝑚²1/√𝐶^∗2𝑏_𝑚, (3.13) for all sufficiently large integers 𝑚, where 𝐶^∗ is a positive number larger than 𝐶. Hence, in view of (3.12) and (3.13), the condition (2.1) is satisfied with 𝜎=1/√𝐶^∗. Moreover, we know that the radius of convergence of power series ∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚 is at least 𝜌 because the convergence radius of the power series expression for 𝑦(𝑥) is at least 𝜌 (see (a) and (3.7)).
According to (3.7) and Theorem 2.1, there exists a solution 𝑦_ℎ(𝑥) of the homogeneous Bessel differential equation (1.3) satisfying (2.3) for all 𝑥∈𝐼_𝜌0. Thus, it follows from (2.2) and (3.12) that 𝑦(𝑥)−𝑦_ℎ(𝑥)=^∞_𝑛=0𝑐_2𝑛𝑥^2𝑛+^∞_𝑛=0𝑐_2𝑛+1𝑥^2𝑛+1=−^∞_𝑛=0𝑥^𝑛2𝑛_𝑖=0𝑎^𝑛_{2𝑖𝑗=𝑖}1𝜈²−2𝑗²−^∞_𝑛=0𝑥^𝑛2𝑛+1_𝑖=0𝑎^𝑛_{2𝑖+1𝑗=𝑖}1𝜈²−2𝑗+1²=^∞_𝑛=01𝜈²−(2𝑛)^2𝑛_𝑖=0𝑎_2𝑖𝑥^{2𝑖𝑛−1}_𝑗=𝑖𝑥²𝜈²−2𝑗²+^∞_𝑛=01𝜈²−(2𝑛+1)^2𝑛_𝑖=0𝑎_2𝑖+1𝑥^{2𝑖+1𝑛−1}_𝑗=𝑖𝑥²𝜈²−2𝑗+1², (3.14) for any 𝑥∈𝐼_𝜌0. Moreover, we have 𝑦(𝑥)−𝑦_ℎ(𝑥)≤^𝜇+1_𝑛=01𝜈²−(2𝑛)^2𝑛_𝑖=0𝑎_2𝑖𝑥^{2𝑖𝑛−1}_𝑗=𝑖𝑥²𝜈²−2𝑗²+^∞_𝑛=𝜇+21𝜈²−(2𝑛)^2𝜇_𝑖=0𝑎_2𝑖𝑥^{2𝑖𝑛−1}_𝑗=𝑖𝑥²𝜈²−2𝑗²+^∞_𝑛=𝜇+21𝜈²−(2𝑛)^2𝑛_𝑖=𝜇+1𝑎_2𝑖𝑥^{2𝑖𝑛−1}_𝑗=𝑖𝑥²𝜈²−2𝑗²+^𝜇+1_𝑛=01𝜈²−(2𝑛+1)^2𝑛_𝑖=0𝑎_2𝑖+1𝑥^{2𝑖+1𝑛−1}_𝑗=𝑖𝑥²𝜈²−2𝑗+1²+^∞_𝑛=𝜇+21𝜈²−(2𝑛+1)^2𝜇_𝑖=0𝑎_2𝑖+1𝑥^{2𝑖+1𝑛−1}_𝑗=𝑖𝑥²𝜈²−2𝑗+1²+^∞_𝑛=𝜇+21𝜈²−(2𝑛+1)^2𝑛_𝑖=𝜇+1𝑎_2𝑖+1𝑥^{2𝑖+1𝑛−1}_𝑗=𝑖𝑥²𝜈²−2𝑗+1², (3.15) for all 𝑥∈𝐼_𝜌0, where 𝜇=[√𝜈²+𝑥²/2].
We know that 𝑥²/|𝜈²−(2𝑗)²|<1 and 𝑥²/|𝜈²−(2𝑗+1)²|<1 for 𝑗≥𝜇+1 and 𝑥²/|𝜈²−(2𝑗)²| or 𝑥²/|𝜈²−(2𝑗+1)²| is not perhaps less than 1 for 𝑗≤𝜇. Then, we have ^𝑛−1_𝑗=𝑖𝑥²𝜈²−2𝑗²=^𝜇_𝑗=𝑖𝑥²𝜈²−2𝑗^2𝑛−1_𝑗=𝜇+1𝑥²𝜈²−2𝑗²≤𝑀_𝑒(𝑥)≤𝑀(𝑥), (3.16) for all 𝑛≥𝜇+2 and 𝑖=0,1,…,𝜇. Similarly, we get ^𝑛−1_𝑗=𝑖𝑥²𝜈²−2𝑗+1²≤𝑀(𝑥), (3.17) for all 𝑛≥𝜇+2 and 𝑖=0,1,…,𝜇.
Thus, it follows from (3.9) and (3.15) that 𝑦(𝑥)−𝑦_ℎ(𝑥)≤^𝜇+1_𝑛=0𝑀(𝑥)𝜈²−(2𝑛)^2𝑛−1_𝑖=0𝑎_2𝑖𝑥^2𝑖+^𝜇+1_𝑛=0𝑎_2𝑛𝑥^2𝑛𝜈²−(2𝑛)²+^∞_𝑛=𝜇+2𝑀(𝑥)𝜈²−(2𝑛)^2𝜇_𝑖=0𝑎_2𝑖𝑥^2𝑖+^∞_𝑛=𝜇+21𝜈²−(2𝑛)^2𝑛_𝑖=𝜇+1𝑎_2𝑖𝑥^2𝑖+^𝜇+1_𝑛=0𝑀(𝑥)𝜈²−(2𝑛+1)^2𝑛−1_𝑖=0𝑎_2𝑖+1𝑥^2𝑖+1+^𝜇+1_𝑛=0𝑎_2𝑛+1𝑥^2𝑛+1𝜈²−(2𝑛+1)²+^∞_𝑛=𝜇+2𝑀(𝑥)𝜈²−(2𝑛+1)^2𝜇_𝑖=0𝑎_2𝑖+1𝑥^2𝑖+1+^∞_𝑛=𝜇+21𝜈²−(2𝑛+1)^2𝑛_𝑖=𝜇+1𝑎_2𝑖+1𝑥^2𝑖+1≤𝑀(𝑥)^∞_𝑛=01𝜈²−(2𝑛)^2𝑛_𝑖=0𝑎_2𝑖𝑥^2𝑖+^∞_𝑛=01𝜈²−(2𝑛+1)^2𝑛_𝑖=0𝑎_2𝑖+1𝑥^2𝑖+1≤𝑀(𝑥)𝐾𝜀^∞_𝑚=01𝜈²−𝑚²≤𝐾𝐿_𝜈𝑀(𝑥)𝜀, (3.18) for 𝑥∈𝐼_𝜌0.
Finally, assume that 𝐶 is sufficiently small and 𝜌 is sufficiently large. Then 𝜌₀ is sufficiently large. For any 𝑗≥0 and 𝑥∈𝐼_𝜌0, we have 𝑥²𝜈²−2𝑗²𝑥≤max²𝜈²−𝑝²,𝑥²𝜈²−𝑝+1². (3.19) If |𝑥| is so large that 1𝜇=2√𝜈²+𝑥²=|𝑥|2≤|𝑥|2, (3.20) then it follows from the definition of 𝑀(𝑥) that 𝑀(𝑥)|𝑥|≤max^|𝑥|+2𝜈²−𝑝^2|𝑥|/2+1,|𝑥|^|𝑥|+2𝜈²−𝑝+1^2|𝑥|/2+1, (3.21) for all sufficiently large |𝑥|.

If 𝜈 is large enough, then we can prove the (local) Hyers-Ulam stability of the Bessel differential equation (1.3) as we see in the following corollary.

Corollary 3.2. Let 𝜈 be a positive nonintegral number and let 𝑝 be a nonnegative integer with 𝑝<𝜈<𝑝+1. Assume that a function 𝑦∈𝒞_𝐾 satisfies the differential inequality (3.3) for all 𝑥∈𝐼_𝜌 and for some 𝜀≥0. Suppose the sequence{𝑏_𝑚} satisfies the condition (3.4) with a Landau constant 𝐶≥0 and define 𝜌₀=min{𝜌,1/√𝐶^∗} for a positive number 𝐶^∗>𝐶. If 𝑥²𝜈²−𝑝²𝑥≤1,²𝜈²−𝑝+1²≤1, (3.22) for all 𝑥∈𝐼_𝜌0, then there exists a solution 𝑦_ℎ(𝑥) of the Bessel differential equation (1.3) such that 𝑦(𝑥)−𝑦_ℎ(𝑥)≤𝐾𝐿_𝜈𝜀, (3.23) for any 𝑥∈𝐼_𝜌0.

Proof. For any 𝑗≥0 and 𝑥∈𝐼_𝜌0, we have 𝑥max²𝜈²−2𝑗²,𝑥²𝜈²−2𝑗+1²𝑥≤max²𝜈²−𝑝²,𝑥²𝜈²−𝑝+1²≤1. (3.24) Thus, we get 𝑀(𝑥)𝑥≤max²𝜈²−𝑝²,𝑥²𝜈²−𝑝+1²,1=1, (3.25) and the assertion is true due to Theorem 3.1.

4. Examples

We will show that there exist functions 𝑦(𝑥) which satisfy all the conditions given in Theorem 3.1 and Corollary 3.2. Let us define a function 𝑦∶𝐼₁₀→ℝ by𝑦(𝑥)=𝐽_100.5(𝑥)+𝑐𝑥²=^∞_𝑚=0𝑏_𝑚𝑥^𝑚, (4.1) where 𝐽_100.5(𝑥) is the Bessel function of the first kind of order 100.5, 𝑛 is a positive integer, and 𝑐 is a constant satisfying𝜀𝑐=999625, (4.2) for some 𝜀>0. It is obvious that the convergence radius of the power series ∑^∞_𝑚=0𝑏_𝑚𝑥^𝑚 is infinity. (So we can set 𝜌=10.) In fact, the infinite series ∑^∞_𝑚=0𝑏_𝑚11^𝑚 converges. So we have limsup_𝑚→∞11^𝑚+1𝑏_𝑚+111^𝑚𝑏_𝑚≤1,limsup_𝑚→∞11^𝑚+2𝑏_𝑚+211^𝑚+1𝑏_𝑚+1≤1. (4.3) Thus, it holds true that limsup_𝑚→∞𝑏_𝑚+2𝑏_𝑚=limsup_𝑚→∞𝑏_𝑚+2𝑏_𝑚+1limsup_𝑚→∞𝑏_𝑚+1𝑏_𝑚≤1121 (4.4) which implies that the sequence {𝑏_𝑚} satisfies the condition (3.4) with a Landau constant 𝐶=1/121. If we take 𝐶^∗=1/100, then 𝜌₀=min{𝜌,1/√𝐶^∗}=10.

Since 𝐽_100.5(𝑥) is a particular solution of the Bessel differential equation (1.3) with 𝜈=100.5, it follows from (4.1) that 𝑥²𝑦^′′(𝑥)+𝑥𝑦^′(𝑥)+𝑥²−404014𝑦(𝑥)=−403854𝑐𝑥²+𝑐𝑥⁴, (4.5) for any 𝑥∈𝐼₁₀.

If we set 𝑎_𝑚=−403854𝑐for𝑚=2,𝑐for𝑚=4,0otherwise, (4.6) then we have 𝑥²𝑦^′′(𝑥)+𝑥𝑦^′(𝑥)+𝑥²−404014𝑦(𝑥)=^∞_𝑚=0𝑎_𝑚𝑥^𝑚,𝑥²𝑦^′′(𝑥)+𝑥𝑦^′(𝑥)+𝑥²−404014𝑦(𝑥)=^∞_𝑚=0𝑎_𝑚𝑥^𝑚=𝑐403854𝑥²−𝑥⁴<999625𝑐=𝜀, (4.7) for all 𝑥∈𝐼₁₀.

Moreover, we have ∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚∑^∞_𝑚=0𝑎_𝑚𝑥^𝑚=(40385/4)𝑐𝑥²−𝑐𝑥⁴(40385/4)𝑐𝑥²+𝑐𝑥⁴>3998540785, (4.8) and hence, we get ^∞_𝑚=0𝑎_𝑚𝑥^𝑚<81577997^∞_𝑚=0𝑎_𝑚𝑥^𝑚, (4.9) for all 𝑥∈𝐼₁₀. That is, {𝑎_𝑚} satisfies the property (b) with 𝐾=8157/7997.

It holds true that 𝑥²𝜈²−𝑝²𝑥<1,²𝜈²−𝑝+1²<1, (4.10) for all 𝑥∈𝐼₁₀, and since 𝐿_100.5=^∞_𝑚=01(𝑚−100.5)²=1(−100.5)²+1(−99.5)²+1(−98.5)²1+⋯+(−0.5)²+10.5²+11.5²+12.5²+13.5²≤1+⋯100²+199²+198²1+⋯+1²+10.5²+10.5²+11²+12²+13²(2)𝜋+⋯≤2𝜁+8=²3+8, (4.11) it follows from Corollary 3.2 that there exists a solution 𝑦_ℎ(𝑥) of the Bessel differential equation (1.3) such that 𝑦(𝑥)−𝑦_ℎ(𝑥)≤8157𝐿7997^100.5𝜀<8157𝜋79973²+8𝜀, (4.12) for any 𝑥∈𝐼₁₀.

Acknowledgment

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (no. 2010-0007143).