Approximation of Analytic Functions by Kummer Functions

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 ∞

m0amxm 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

anxynx an−1xyn−1x · · ·a1xyx a0xyx hx≤ε 1.1

for allx∈Iand for someε≥0, there exists a solutionf0:I → Y of the differential equation

anxynx an−1xyn−1x · · ·a1xyx a0xyx hx 0 1.2

|y_{x−yx| ≤ε, whereIis an open subinterval ofR, then there exists a solutionf}

0 :I → R of the differential equationy_{x yxsuch that|fx−f}

0x| ≤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−x2yx−2xyx pp1yx

∞

m0

amxm 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

xy_{x β−xyx−αyx} ∞

m0

amxm, 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

xy_{x β−xyx−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

Mα, β, x∞

m0 α_m

m!β_mx

m_{, 2.2}

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

Let us define

Uα, β, x π

sinβπ

Mα, β, x

Γ1α−βΓβ −x 1−βM

1α−β,2−β, x ΓαΓ2−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 factorx1−β_{in the above definition ofUα, β, x.}

By considering this fact, we define

Iρ

⎧ ⎨ ⎩

−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 neitherαnor1α−βis a nonpositive integer. Assume that the radius of convergence of the power series∞_m0a_mxmisρ >0and that there exists a real numberμ≥0with

m−1!β_mam

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_mβ_iai

m!α_i1β_mxm, 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_mβ_iai

m−1!α_i1

β_mxm−

1∞

m0

m

i0

i!α_m1

β_iai

m!α_i1

β_m1xm,

y

px

∞

m1

m

i0

i!α_m1

β_iai

m−1!α_i1

β_m1xm−1,

we have

xy

px β−xypx−αypx a0 ∞

m1

m

i0

i!α_m1

β_imβai

m!α_i1

β_m1 xm

−∞

m1

m−1

i0

i!α_mβ_imαai

m!α_i1β_m xm

a0 ∞

m1

amxm,

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_mβ_iai

m!α_i1β_mxm ∞

m1

cmxm. 2.9

Then, it follows from2.5that

lim

m→ ∞

cm1

cm

≤ lim

m→ ∞

α_βm_m

⎡ ⎣ 1

m1

m

m1

m−1!β_mam

α_m1

m−1

i0

i!β_iai

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 ypx. 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−αypxhas the same convergence region as that ofypx. This implies that

ypx is well defined on Iρ0 and so does for yx in2.6 becauseyhx 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 , am1

4m2_−6m−3

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

m3m−1 am−1·

3m7 3m ·

am

am−1 ·

m

m1

m−1!

β_m−1am−1

α_m ·

3m7 3m ·

am

am−1 ·

m

m1

≤μm−1!

β_m−1am−1 α_m

≤μm−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∞_m0bmxmwhose radius of convergence is at

leastρ;

bit holds true that ∞_m0|amxm| ≤ K|m∞0amxm|for allx ∈ Iρ, where am m

βm1bm1−mαbmfor eachm∈N0.

It should be remarked that the power series∞_m0a_mxminbhas the same radius of convergence as that of∞_m0bmxmgiven 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 neitherαnor1α−βis a nonpositive integer. Suppose a functiony:Iρ → Cis representable by a power series∞m0bmxmwhose radius of convergence is at leastρ >0. Assume that there exist nonnegative constantsμ /0andνsatisfying the condition

m−1!β_mam

α_m1 ≤μ

m−1

i0

i!β_iai

α_i1 ≤ν

m1!β_mam

α_m1

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−xyx−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−yhx≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

ν μ·

2α−1

α Kε forα >1,

ν μ

m0−1

m0

_mm_α1− m2

m1α

m01

m0α

Kε forα≤1,

3.3

for anyx∈I_ρ0, wherem0max{0, −α}.

Proof. By the definition ofam, we have

xy_{x β−xyx−αyx}

∞

m0

mβm1bm1−mαbmxm

∞

m0

amxm

3.4

for allx∈Iρ. So by3.2we have

∞

m0

amxm

≤ε 3.5

for anyx∈Iρ0. Sincey∈ CK, this inequality together withbyields

∞

m0

|amxm| ≤K

∞

m0

amxm

≤Kε 3.6

for eachx∈Iρ0.

By Abel’s formulasee18, Theorem 6.30, we have

n

m0

|amxm|m_mα1

_n

i0 aixi

_nn₁2_α

n

m0 _m

i0 aixi

m_mα1−_mm₁2_α

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 yhx

∞

m0

m−1

i0

i!α_mβ_iai

m!α_i1

β_mxm 3.9

for allx∈Iρ0. By using3.1,3.6,3.7, and3.8, we can estimate

_yx−yhx≤ ∞

m0

amxmm_mα1

_m1!αm_β1_mam

m−1

i0

i!β_ia_i α_i1

≤ ν

μnlim→ ∞

n

m0

|amxm|_mm_α1

≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ν

μnlim→ ∞ Kε

_nn₁2_αn

m0

Kε

_m 2

m1α−

m1

mα

forα >1,

ν

μnlim→ ∞ Kε

_nn₁2_αm0−1

m0

Kεm1

mα

− m2

m1α

n

mm0Kε

_m 1

mα−

m2

m1α

forα≤1

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ν μ·

2α−1

α Kε forα >1,

ν μ

m0−1

m0

m_mα1−_mm2

1α

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 solutionyhxof the Kummer’s equation on a largerpunctured

interval.

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

Moreover, assume that

lim

m→ ∞

m−1!β_mam

α_m1

0, 0<

∞

i0

i!β_iai

α_i1

<∞ 3.11

and there exists a nonnegative constantνsatisfying

m−1

i0

i!β_iai

α_i1 ≤ν

m1!β_mam

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−ynx≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

ν·2α−1

α Kε forα >1,

ν m0−1

m0

_mm_α1− 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 ≤ νn

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

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

yx ∞

m0

bmxm 4.2

for anyx∈Iρ.

Then, we seta_m mβm1b_m1−mαbm, that is,

a0

10 3 ·

ε

s, am

14m

2_−6m−3 3m2_m1

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

s

41 36·

ε

s a1a3, 4.4

we get

∞

m0

amxm

a0a1xa2x2a3x3

a4x4a5x5

a6x6a7x7

· · ·

≥a0a1xa2x2a3x3

≥a0−a1−a3

73 36·

ε s

4.5

for eachx∈Iρand

∞

m0

|amxm| ≤

∞

m0

amρm≤

10

3

∞

m1 5 3ρ

m

ε

s ε 4.6

for allx∈I_ρ. Hence, we obtain

∞

m0

|amxm| ≤K

∞

m0

amxm

4.7

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_mam

α_m1 ≥

10·13·16· · ·3m7

3m ·

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 3m

m−1

i1

3m−i

3i10· · ·3m7· 1

i1 · 1 6

ε s

≤10 1 3

10·13·16· · ·3m7 3m

m−1

i1 1 i12 ·

1 6

ε s

≤1010·13·16· · ·3m7 3m

1 10

1

6ζ2−1 _ε

s

5π2−12

3 ·

10·13·16· · ·3m7

3m ·

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−xyx−αyx}

∞

m0

amxm

≤

∞

m0

|amxm| ≤ε 4.10

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

equation2.1such that

_yx−yhx≤ 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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