2. General Solution of 1.4

Volume 2010, Article ID 898274,11pages doi:10.1155/2010/898274

Research Article

Approximation of Analytic Functions by Kummer Functions

Soon-Mo Jung

Mathematics Section, College of Science and Technology, Hongik University, Jochiwon 339-701, Republic of Korea

Correspondence should be addressed to Soon-Mo Jung,[email protected] Received 3 February 2010; Revised 27 March 2010; Accepted 31 March 2010 Academic Editor: Alberto Cabada

Copyrightq2010 Soon-Mo Jung. 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.

We solve the inhomogeneous Kummer differential equation of the formxy β−xy−αy

_∞

m0amx^m 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 thatX andY are a topological vector space and a normed space, respectively, and thatIis an open subset ofX. If for any functionf:I → Ysatisfying the differential inequality a_nxyⁿx a_n−1xyⁿ⁻¹x · · ·a1xyx a0xyx hx≤ε 1.1

for allx∈Iand for someε≥0, there exists a solutionf0:I → Y of the differential equation a_nxyⁿx a_n−1xyⁿ⁻¹x · · ·a1xyx a0xyx hx 0 1.2

such thatfx−f0x ≤Kεfor anyx∈I, whereKεdepends onεonly, then we say that the above differential equation satisfies the Hyers-Ulam stability or the local Hyers-Ulam stability if the domainI is not the whole spaceX. We may apply this terminology for other differential equations. For more detailed definition of the Hyers-Ulam stability, refer to1–6.

Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equationssee7,8. Here, we will introduce a result of Alsina and Ger see9. If a differentiable functionf : I → Ris a solution of the differential inequality

|yx−yx| ≤ε, whereIis an open subinterval ofR, then there exists a solutionf0 :I → R of the differential equationyx yxsuch that|fx−f0x| ≤3εfor anyx∈I.

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 equationyx λyx see also11.

Using the conventional power series method, the author12investigated the general solution of the inhomogeneous Legendre differential equation of the form

1−x²

yx−2xyx p p1

yx ^∞

m0

a_mx^m 1.3

under some specific conditions, wherepis 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 0 by the Legendre function with an error bound expressed byCx²/1−x² see13–16.

In Section 2 of this paper, employing power series method, we will determine the general solution of the inhomogeneous Kummerdifferentialequation

xyx β−x

yx−αyx ^∞

m0

a_mx^m, 1.4

whereαandβare constants and the coefficientsamof the power series are given such that the radius of convergence isρ > 0, whose value is in general permitted to be infinite. Moreover, using the idea from12,13,15, we will prove the Hyers-Ulam stability of the Kummer’s equation in a class of special analytic functionssee the classCKinSection 3.

In this paper,N0 and Zdenote the set of all nonnegative integers and the set of all integers, respectively. For each real numberα, we use the notation αto denote the ceiling of α, that is, the least integer not less thanα.

2. General Solution of 1.4

The Kummerdifferentialequation xyx

β−x

yx−αyx 0, 2.1

which is also called the confluent hypergeometric differential equation, appears frequently in practical problems and applications. The Kummer’s equation2.1has a regular singularity atx0 and an irregular singularity at∞. A power series solution of2.1is given by

M α, β, x

^∞

m0

α_m m!

β

m

x^m, 2.2

whereα_mis the factorial function defined byα₀1 andα_mαα1α2· · ·αm−1 for allm∈N. The above power series solution is called the Kummer function or the confluent

hypergeometric function. We know that if neitherαnorβis a nonpositive integer, then the power series forMα, β, xconverges for all values ofx.

Let us define

U α, β, x

π sinβπ

M α, β, x Γ

1α−β Γ

β −x^1−βM

1α−β,2−β, x ΓαΓ

2−β

. 2.3

We know that if β /1 then Mα, β, x and Uα, β, x are independent solutions of the Kummer’s equation 2.1. When β > 1, Uα, β, x is not defined at x 0 because of the factorx^1−βin the above definition ofUα, β, x.

By considering this fact, we define

I_ρ

⎧⎨

⎩ −ρ, ρ

,

forβ <1 , −ρ,0

∪ 0, ρ

,

forβ >1

, 2.4

for any 0 < ρ ≤ ∞. It should be remarked that if β /∈Z and bothαand 1α−β are not nonpositive integers, thenMα, β, xandUα, β, xconverge for allx∈I_∞see17, Section 13.1.3.

Theorem 2.1. Letαandβbe real constants such thatβ /∈Zand neitherαnor 1α−βis a nonpositive integer. Assume that the radius of convergence of the power series_∞

m0a_mx^misρ >0 and that there exists a real numberμ≥0 with

m−1!

β

ma_m α_m1

≤μ

m−1

i0

i!

β

ia_i α_i1

2.5

for all sufficiently large integers m. Let us define ρ0 min{ρ,1/μ} and 1/0 ∞. Then, every solutiony:Iρ0 → Cof the inhomogeneous Kummer’s equation1.4can be represented by

yx yhx ^∞

m1 m−1

i0

i!α_m β

iai

m!α_i1 β

m

x^m, 2.6

whereyhxis a solution of the Kummer’s equation2.1.

Proof. Assume that a functiony :I_ρ0 → Cis given by2.6. We first prove that the function ypx, defined byyx−yhx, satisfies the inhomogeneous Kummer’s equation1.4. Since

y_px ^∞

m1 m−1

i0

i!α_m β

iai

m−1!α_i1 β

m

x^m−1∞

m0

m i0

i!α_m1 β

iai

m!α_i1 β

m1

x^m,

y_px ^∞

m1

m i0

i!α_m1 β

ia_i m−1!α_i1

β

m1

x^m−1,

2.7

we have

xy_px β−x

y_px−αy_px a0^∞

m1

m i0

i!α_m1 β

i

mβ a_i m!α_i1

β

m1

x^m

−^∞

m1 m−1

i0

i!α_m β

imαai

m!α_i1 β

m

x^m a0^∞

m1

amx^m,

2.8

which proves thatypxis a particular solution of the inhomogeneous Kummer’s equation 1.4.

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

ypx ^∞

m1 m−1

i0

i!α_m β

iai

m!α_i1 β

m

x^m∞

m1

cmx^m. 2.9

Then, it follows from2.5that

mlim→ ∞

c_m1 cm

≤ lim

m→ ∞

αm βm

⎡

⎣ 1

m1 m m1

m−1!

β

ma_m α_m1

m−1

i0

i!

β

ia_i α_i1

−1⎤

⎦

≤μ.

2.10

Therefore, the power series expression of ypx converges for all x ∈ I1/μ. Moreover, the convergence region of the power series for ypx is the same as those of power series for y_pxand y_px. In this paper, the convergence region will denote the maximum open set where the relevant power series converges. Hence, the power series expression forxy_px β−xy_px−αy_pxhas the same convergence region as that ofy_px. This implies that y_px is well defined on I_ρ0 and so does for yx in2.6 becausey_hx converges for all x∈I∞under our hypotheses forαandβsee aboveTheorem 2.1.

Since every solution to 1.4 can be expressed as a sum of a solution yhx of the homogeneous equation and a particular solutiony_pxof the inhomogeneous equation, every solution of1.4is certainly in the form of2.6.

Remark 2.2. We fixα1 andβ10/3,and we define

a0 10

3 , a_m14m²−6m−3

3m²m1 2.11

for everym∈N. Then, since limm→ ∞am/am−11, there exists a real numberμ >1 such that

m−1!

β

mam

α_m1

10·13·16· · ·3m4

m3^m−1 a_m−1·3m7 3m · am

a_m−1 · m m1 m−1!

β

m−1a_m−1

α_m ·3m7 3m · a_m

am−1 · m m1

≤μm−1!

β

m−1am−1

α_m

≤μ

m−1

i0

i!

β

iai

α_i1

2.12

for all sufficiently large integersm. Hence, the sequence{a_m}satisfies condition2.5for all sufficiently large integersm.

3. Hyers-Ulam Stability of 2.1

In this section, letαandβbe real constants and assume thatρis a constant with 0< ρ ≤ ∞.

For a givenK≥0, let us denoteC_Kthe set of all functionsy:I_ρ → Cwith the propertiesa andb:

ayxis represented by a power series_∞

m0bmx^mwhose radius of convergence is at leastρ;

bit holds true that _∞

m0|amx^m| ≤ K|_∞

m0amx^m|for allx ∈ Iρ, where am m βm1b_m1−mαb_mfor eachm∈N0.

It should be remarked that the power series_∞

m0a_mx^minbhas the same radius of convergence as that of_∞

m0bmx^mgiven ina.

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 asMα, β, xandM1α−β,2−β, x see the first part ofSection 2.

Theorem 3.1. Letαandβbe real constants such thatβ /∈Zand neitherαnor 1α−βis a nonpositive integer. Suppose a functiony:Iρ → Cis representable by a power series_∞

m0bmx^mwhose radius of convergence is at leastρ >0. Assume that there exist nonnegative constantsμ /0 andνsatisfying the condition

m−1!

β

mam

α_m1 ≤μ

m−1

i0

i!

β

iai

α_i1 ≤ν

m1!

β

mam

α_m1

3.1

for allm ∈ N0, wheream mβm1bm1−mαbm. Indeed, it is sufficient for the first inequality in3.1to hold true for all sufficiently large integersm. Let us defineρ0min{ρ,1/μ}. If y∈ CKand it satisfies the differential inequality

xyx β−x

yx−αyx≤ε 3.2

for allx∈I_ρ0and for someε≥0, then there exists a solutiony_h:I_∞ → Cof the Kummer’s equation 2.1such that

yx−y_hx≤

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ ν

μ·2α−1

α Kε forα >1,

ν μ

m0−1 m0

m1

mα −

m2 m1α

m01 m0α

Kε forα≤1,

3.3

for anyx∈I_ρ0, wherem0max{0, −α}.

Proof. By the definition ofam, we have xyx

β−x

yx−αyx ^∞

m0

mβ

m1b_m1−mαb_m x^{m ∞}

m0

a_mx^m

3.4

for allx∈Iρ. So by3.2we have

∞ m0

amx^m

≤ε 3.5

for anyx∈I_ρ0. Sincey∈ C_K, this inequality together withbyields ∞

m0

|a_mx^m| ≤K

∞ m0

a_mx^m

≤Kε 3.6

for eachx∈Iρ0.

By Abel’s formulasee18, Theorem 6.30, we have n

m0

|amx^m| m1

mα

_n

i0

aixⁱ

n2 n1α

ⁿ

m0

_m

i0

aixⁱ m1

mα −

m2 m1α

3.7

for anyx∈Iρ0andn∈N. Withm0max{0, −α} −αis the ceiling of−α, we know that

if α >1, then m1

mα < m2

m1α form≥0;

if α≤1, then m1

mα ≥ m2

m1α form≥m0.

3.8

Due to3.4, it follows fromTheorem 2.1and2.6that there exists a solutiony_hxof the Kummer’s equation2.1such that

yx y_hx ^∞

m0 m−1

i0

i!α_m β

ia_i m!α_i1

β

m

x^m 3.9

for allx∈Iρ0. By using3.1,3.6,3.7, and3.8, we can estimate yx−y_hx≤ ^∞

m0

a_mx^mm1 mα

α_m1 m1!

β

mam

m−1

i0

i!

β

ia_i α_i1

≤ ν μ lim

n→ ∞

n m0

|a_mx^m| m1

mα

≤

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩ ν μlim

n→ ∞ Kε n2

n1α ⁿ

m0

Kε

m2

m1α− m1 mα

forα >1,

ν μlim

n→ ∞ Kε n2

n1α ^m0−1

m0

Kε

m1 mα

− m2

m1α

ⁿ

mm0

Kε m1

mα− m2 m1α

forα≤1

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ ν

μ·2α−1

α Kε forα >1,

ν μ

m0−1 m0

m1

mα −

m2 m1α

m01 m0α

Kε forα≤1

3.10

for allx∈Iρ0.

We now assume a stronger condition, in comparison with3.1, to approximate the given functionyxby a solutiony_hxof the Kummer’s equation on a largerpunctured interval.

Corollary 3.2. Letαandβbe real constants such thatβ /∈Zand neitherαnor 1α−βis a nonpositive integer. Suppose a function y : I_∞ → C is representable by a power series _∞

m0b_mx^m which

converges for allx∈ I∞. For everym∈ N0, let us defineam mβm1bm1−mαbm. Moreover, assume that

mlim→ ∞

m−1!

β

ma_m

α_m1 0, 0<

∞

i0

i!

β

iai

α_i1

<∞ 3.11

and there exists a nonnegative constantνsatisfying

m−1

i0

i!

β

ia_i α_i1

≤ν

m1!

β

ma_m α_m1

3.12

for allm∈N0. Ify ∈ CKand it satisfies the differential inequality3.2for allx∈I∞and for some ε≥0, then there exists a solutiony_n:I_∞ → Cof the Kummer’s equation2.1such that

yx−y_nx≤

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

ν·2α−1

α Kε forα >1,

ν ^m0−1

m0

m1

mα −

m2 m1α

m01 m0α

Kε forα≤1

3.13

for anyx∈In, wherem0max{0, −α}andnis a sufficiently large integer.

Proof. In view of3.11and3.12, we can choose a sufficiently large integernwith

m−1!

β

mam

α_m1 ≤ 1

n

m−1

i0

i!

β

iai

α_i1 ≤ ν

n

m1!

β

mam

α_m1

, 3.14

where the first inequality holds true for all sufficiently largem,and the second one holds true for allm∈N0.

If we defineρ0n, thenTheorem 3.1implies that there exists a solutionyn :I∞ → C of the Kummer’s equation such that the inequality given for|yx−y_nx|holds true for any x∈In.

4. An Example

We fixα1,β10/3, ε >0, and 0< ρ <1. And we define

b00, b_m ε s · 1

m² 4.1

for allm∈N, where we sets 5/32−ρ/1−ρ. We further define

yx ^∞

m0

bmx^m 4.2

for anyx∈I_ρ.

Then, we seta_m mβm1b_m1−mαb_m, that is,

a0 10 3 ·ε

s, a_m

14m²−6m−3 3m²m1

ε s ≤ 5

3·ε

s 4.3

for everym∈N. Obviously, alla_ms are positive, and the sequence{a_m}is strictly monotone decreasing, from the 4th term on, to ε/s. More precisely, a0 > a1 < a2 < a3 < a4 > a5 >

a6>· · ·. Since

a0 10 3 ·ε

s > 1 6 ·ε

s41 36·ε

s a1a3, 4.4

we get

∞ m0

amx^m

a0a1xa2x²a3x³

a4x⁴a5x⁵

a6x⁶a7x⁷ · · ·

≥a0a1xa2x²a3x³

≥a0−a1−a3

73 36·ε

s

4.5

for eachx∈Iρand ∞ m0

|a_mx^m| ≤^∞

m0

a_mρ^m≤ 10

3 ^∞

m1

5 3ρ^m

ε

s ε 4.6

for allx∈I_ρ. Hence, we obtain ∞ m0

|a_mx^m| ≤K

∞ m0

a_mx^m

4.7 for anyx∈I_ρ, whereK 60/73·2−ρ/1−ρ, implying thaty∈ C_K.

We will now show that{am}satisfies condition3.1. For anym∈N, we have

m−1

i0

i!

β

iai

α_i1

a0^m−1

i1

10·13·16· · ·3i7 i13ⁱ ai

≤ 10 3 ^m−1

i1

10·13·16· · ·3i7 i13ⁱ ·5

3 ε

s,

m1!

β

ma_m α_m1

≥ 10·13·16· · ·3m7

3^m ·1

6·ε s,

4.8

since lim_{m→ ∞}a_mε/s.

It follows from4.8that

m−1

i0

i!

β

iai

α_i1 ≤10 1

3 ^m−1

i1

10·13·16· · ·3i7 i13ⁱ ·1

6 ε

s 10 1

3 10·13· · ·3m7 3^m

m−1

i1

3^m−i

3i10· · ·3m7· 1 i1 ·1

6 ε

s

≤10 1

3 10·13·16· · ·3m7 3^m

m−1

i1

1 i1² ·1

6 ε

s

≤1010·13·16· · ·3m7 3^m

1 101

6ζ2−1 ε

s 5π²−12

3 ·10·13·16· · ·3m7

3^m ·1

6·ε s

≤ 5π²−12 3

m1!

β

mam

α_m1 .

4.9

We know that the inequality4.9is also true form0.

On the other hand, in view ofRemark 2.2, there exists a constant μ > 1 such that inequality 2.12 holds true for all sufficiently large integers m. By 2.12 and 4.9, we conclude that{a_m}satisfies condition3.1withν 5π²−12μ/3.

Finally, it follows from4.6that

xyx β−x

yx−αyx

∞ m0

amx^m ≤^∞

m0

|amx^m| ≤ε 4.10

for allx∈I_ρ0withρ0min{ρ,1/μ}.

According to Theorem 3.1, there exists a solution yh : I∞ → C of the Kummer’s equation2.1such that

yx−y_hx≤ 100π²−240 73 ·2−ρ

1−ρε 4.11

for allx∈Iρ0.

Acknowledgments

The author would like to express his cordial thanks to the referee for his/her useful comments. This work was supported by National Research Foundation of Korea Grant funded by the Korean GovernmentNo. 2009-0071206.

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