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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 898274, 11 pages
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,
Received 3 February 2010; Revised 27 March 2010; Accepted 31 March 2010
Academic Editor: Alberto Cabada
Copyright q 2010 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.
β − x y − αy
We solve the inhomogeneous Kummer differential equation of the form xy
∞
m
and apply this result to the proof of a local Hyers-Ulam stability of the Kummer
m 0 am x
differential equation in a special class of analytic functions.
1. Introduction
Assume that X and Y are a topological vector space and a normed space, respectively, and
that I is an open subset of X. If for any function f : I → Y satisfying the differential inequality
an x y
n
x
an−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
an x y
n
x
an−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 X . 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 f : I → R is a solution of the differential inequality
2
Journal of Inequalities and Applications
|y x − y x | ≤ ε, where I is an open subinterval of R, then there exists a solution f0 : I → R
y x such that |f x − f0 x | ≤ 3ε for any x ∈ I.
of the differential equation y x
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
λy x see also 11 .
equation y x
Using the conventional power series method, the author 12 investigated the general
solution of the inhomogeneous Legendre differential equation of the form
1 − x2 y x − 2xy x
p p
∞
1 y x
am xm
1.3
m 0
under some specific conditions, where p 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 0 by the Legendre function with an error bound
see 13–16 .
expressed by C x2 / 1 − x2
In Section 2 of this paper, employing power series method, we will determine the
general solution of the inhomogeneous Kummer differential equation
xy x
∞
β − x y x − αy x
am xm ,
1.4
m 0
where α and β are constants and the coefficients am of the power series are given such that the
radius of convergence is ρ > 0, 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 CK in Section 3 .
In this paper, N0 and Z denote the set of all nonnegative integers and the set of all
integers, respectively. For each real number α, we use the notation α to denote the ceiling of
α, that is, the least integer not less than α.
2. General Solution of 1.4
The Kummer differential equation
xy x
β − x y x − αy x
0,
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 x 0 and an irregular singularity at ∞. A power series solution of 2.1 is given by
M α, β, x
∞
m
α m m
x ,
0 m! β m
2.2
where α m is the factorial function defined by α 0 1 and α m α α 1 α 2 · · · α m−1
for all m ∈ N. The above power series solution is called the Kummer function or the confluent
Journal of Inequalities and Applications
3
hypergeometric function. We know that if neither α nor β is a nonpositive integer, then the
power series for M α, β, x converges for all values of x.
Let us define
U α, β, x
M 1 α − β, 2 − β, x
M α, β, x
π
− x1−β
sin βπ Γ 1 α − β Γ β
Γ α Γ 2−β
.
2.3
We know that if β / 1 then M α, β, x and U α, β, x are independent solutions of the
Kummer’s equation 2.1 . When β > 1, U α, β, x is not defined at x
0 because of the
factor x1−β in the above definition of U α, β, x .
By considering this fact, we define
⎧
⎨ −ρ, ρ ,
for β < 1 ,
⎩ −ρ, 0 ∪ 0, ρ ,
Iρ
for β > 1 ,
2.4
for any 0 < ρ ≤ ∞. It should be remarked that if β / Z and both α and 1 α − β are not
∈
nonpositive integers, then M α, β, x and U α, β, x converge for all x ∈ I∞ see 17, Section
13.1.3 .
Theorem 2.1. Let α and β be real constants such that β / Z and neither α nor 1 α−β is a nonpositive
∈
integer. Assume that the radius of convergence of the power series ∞ 0 am xm is ρ > 0 and that there
m
exists a real number μ ≥ 0 with
m−1 ! β
α
a
m m
m 1
≤μ
m−1 i!
β i ai
α i 1
i 0
2.5
for all sufficiently large integers m. Let us define ρ0
min{ρ, 1/μ} and 1/0
∞. Then, every
solution y : Iρ0 → C of the inhomogeneous Kummer’s equation 1.4 can be represented by
y x
∞ m−1
yh x
m 1 i 0
i! α
m
m! α
β i ai
i 1
β
xm ,
2.6
m
where yh x is a solution of the Kummer’s equation 2.1 .
Proof. Assume that a function y : Iρ0 → C is given by 2.6 . We first prove that the function
yp x , defined by y x − yh x , satisfies the inhomogeneous Kummer’s equation 1.4 . Since
yp x
∞ m−1
m 1i 0
i! α
m
m−1 ! α
yp x
β i ai
i 1
∞ m
m 1i 0
β
∞ m
xm−1
m 0i 0
m
i! α
m 1
m−1 ! α
i! α
m 1
m! α
i 1
β i ai
i 1
β
xm−1 ,
m 1
β i ai
β
m 1
xm ,
2.7
4
Journal of Inequalities and Applications
we have
xyp x
β − x yp x − αyp x
∞ m
a0
i! α
∞ m−1 i!
m 1i 0
∞
a0
β
m! α
m 1i 0
−
m 1
α
m
β
m! α
i
i
m
β ai
i 1
β
m
α ai
i 1
β
xm
m 1
xm
2.8
m
am xm ,
m 1
which proves that yp x is a particular solution of the inhomogeneous Kummer’s equation
1.4 .
We now apply the ratio test to the power series expression of yp x as follows:
yp x
∞ m−1
m 1i 0
i! α
m! α
m
β i ai
i 1
β
m
xm
∞
cm x m .
2.9
m 1
Then, it follows from 2.5 that
cm 1
α
≤ lim
lim
m → ∞ cm
m→∞ β
⎡
m ⎣ 1
m m 1
m
m 1
m − 1 ! β m am
α m 1
m−1 i!
i 0
β i ai
α i 1
−1
⎤
⎦
2.10
≤ μ.
Therefore, the power series expression of yp x converges for all x ∈ I1/μ . Moreover, the
convergence region of the power series for yp x is the same as those of power series for
yp x and yp x . In this paper, the convergence region will denote the maximum open set
where the relevant power series converges. Hence, the power series expression for xy p x
β − x yp x − αyp x has the same convergence region as that of yp x . This implies that
yp x is well defined on Iρ0 and so does for y x in 2.6 because yh x converges for all
x ∈ I∞ under our hypotheses for α and β see above Theorem 2.1 .
Since every solution to 1.4 can be expressed as a sum of a solution yh x of the
homogeneous equation and a particular solution yp x of the inhomogeneous equation, every
solution of 1.4 is certainly in the form of 2.6 .
Remark 2.2. We fix α
1 and β
a0
10/3, and we define
10
,
3
am
1
4m2 − 6m − 3
3m2 m 1
2.11
Journal of Inequalities and Applications
5
for every m ∈ N. Then, since limm → ∞ am /am−1
m−1 ! β
α
a
m m
1, there exists a real number μ > 1 such that
10 · 13 · 16 · · · 3m
m3m−1
m 1
a
m−1 m−1
m−1 ! β
α
≤μ
≤μ
m
m−1 i!
i 0
·
am−1 ·
3m 7 am
m
·
·
3m
am−1 m 1
3m 7 am
m
·
·
3m
am−1 m 1
a
m−1 m−1
m−1 ! β
α
4
2.12
m
β i ai
α i 1
for all sufficiently large integers m. Hence, the sequence {am } satisfies condition 2.5 for all
sufficiently large integers m.
3. Hyers-Ulam Stability of 2.1
In this section, let α and β be real constants and assume that ρ is a constant with 0 < ρ ≤ ∞.
For a given K ≥ 0, let us denote CK the set of all functions y : Iρ → C with the properties a
and b :
a y x is represented by a power series
least ρ;
b it holds true that
β m 1 bm 1 − m
∞
m 0
bm xm whose radius of convergence is at
∞
m 0
|am xm | ≤ K| ∞ 0 am xm | for all x ∈ Iρ , where am
m
α bm for each m ∈ N0 .
m
It should be remarked that the power series ∞ 0 am xm in b has the same radius of
m
convergence as that of ∞ 0 bm xm given in a .
m
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 M α, β, x and M 1 α − β, 2 − β, x see the
first part of Section 2 .
Theorem 3.1. Let α and β be real constants such that β / Z and neither α nor 1 α−β is a nonpositive
∈
integer. Suppose a function y : Iρ → C is representable by a power series ∞ 0 bm xm whose radius
m
of convergence is at least ρ > 0. Assume that there exist nonnegative constants μ / 0 and ν satisfying
the condition
m−1 ! β
α
m 1
a
m m
≤μ
m−1 i!
i 0
m
β i ai
≤ν
α i 1
1! β
α
m 1
a
m m
3.1
6
Journal of Inequalities and Applications
m β m 1 bm 1 − m α bm . Indeed, it is sufficient for the first
for all m ∈ N0 , where am
inequality in 3.1 to hold true for all sufficiently large integers m. Let us define ρ0 min{ρ, 1/μ}. If
y ∈ CK and it satisfies the differential inequality
xy x
≤ε
β − x y x − αy x
3.2
for all x ∈ Iρ0 and for some ε ≥ 0, then there exists a solution yh : I∞ → C of the Kummer’s equation
2.1 such that
≤
y x − yh x
⎧ ν 2α − 1
⎪ ·
Kε
⎪
⎪μ
⎪
α
⎨
⎪ν
⎪
⎪
⎪
⎩
μ
for any x ∈ Iρ0 , where m0
m0 −1
m
m
m 0
for α > 1 ,
m 2
1
−
α
m 1 α
m0
m0
1
Kε
α
3.3
for α ≤ 1 ,
max{0, −α }.
Proof. By the definition of am , we have
xy x
β − x y x − αy x
∞
m
β m
1 bm
1
α bm xm
− m
m 0
∞
3.4
am xm
m 0
for all x ∈ Iρ . So by 3.2 we have
∞
am xm ≤ ε
3.5
m 0
for any x ∈ Iρ0 . Since y ∈ CK , this inequality together with b yields
∞
|am xm | ≤ K
m 0
∞
am xm ≤ Kε
3.6
m 0
for each x ∈ Iρ0 .
By Abel’s formula see 18, Theorem 6.30 , we have
n
|am xm |
m 0
m
m
n
i 0
1
α
ai xi
n
n
n
2
1
α
m
m 0
i 0
ai xi
m
m
m 2
1
−
α
m 1 α
3.7
Journal of Inequalities and Applications
for any x ∈ Iρ0 and n ∈ N. With m0
7
max{0, −α }
−α is the ceiling of −α , we know that
m
m
m 2
1
<
α m 1 α
for m ≥ 0;
m
if α ≤ 1, then
m
1
m 2
≥
α m 1 α
for m ≥ m0 .
if α > 1, then
3.8
Due to 3.4 , it follows from Theorem 2.1 and 2.6 that there exists a solution yh x of
the Kummer’s equation 2.1 such that
y x
yh x
∞ m−1
i! α
m
m! α
m 0 i 0
β i ai
i 1
β
xm
3.9
m
for all x ∈ Iρ0 . By using 3.1 , 3.6 , 3.7 , and 3.8 , we can estimate
y x − yh x
≤
∞
am xm
m 0
m
m
1
α
m
αm 1
1 ! β m am
m−1 i!
i 0
β i ai
α i 1
n
ν
m 1
lim
|am xm |
μ n → ∞m 0
m α
⎧
⎪ν
⎪ lim Kε n 2
⎪
⎪
⎪μn→∞
⎪
n 1 α
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
≤ ν lim Kε n 2
⎪μn→∞
⎪
n 1 α
⎪
⎪
⎪
⎪
⎪
⎪
n
⎪
⎪
m
⎪
⎪
Kε
⎪
⎩
m
m m0
≤
n
Kε
m 0
m0 −1
Kε
m 0
m
m 2
−
m 1 α m
m
m
m0 −1
m 0
m
m
for α > 1 ,
m 2
1
−
α
m 1 α
m 2
1
−
α m 1 α
⎧ ν 2α − 1
⎪ ·
Kε
⎪
⎪μ
⎪
α
⎨
⎪ν
⎪
⎪
⎪
⎩
μ
1
α
for α ≤ 1
for α > 1 ,
m 2
1
−
α
m 1 α
m0
m0
1
Kε
α
for α ≤ 1
3.10
for all x ∈ Iρ0 .
We now assume a stronger condition, in comparison with 3.1 , to approximate the
given function y x by a solution yh x of the Kummer’s equation on a larger punctured
interval.
Corollary 3.2. Let α and β be real constants such that β / Z and neither α nor 1 α−β is a nonpositive
∈
integer. Suppose a function y : I∞ → C is representable by a power series ∞ 0 bm xm which
m
8
Journal of Inequalities and Applications
converges for all x ∈ I∞ . For every m ∈ N0 , let us define am
Moreover, assume that
lim
m→∞
m − 1 ! β m am
α m 1
0,
0<
m
β m
1 bm
1
− m
α bm .
∞
i
i! β i ai
<∞
α i 1
0
3.11
and there exists a nonnegative constant ν satisfying
m−1 i!
m
β i ai
≤ν
αi 1
i 0
1! β
α
a
m m
3.12
m 1
for all m ∈ N0 . If y ∈ CK and it satisfies the differential inequality 3.2 for all x ∈ I∞ and for some
ε ≥ 0, then there exists a solution yn : I∞ → C of the Kummer’s equation 2.1 such that
y x − yn x
≤
⎧
⎪ν · 2α − 1 Kε
⎪
⎪
⎪
⎨
α
⎪
⎪
⎪ν
⎪
⎩
for any x ∈ In , where m0
m0 −1
m 0
for α > 1 ,
m 2
1
−
α
m 1 α
m
m
m0
m0
3.13
1
Kε
α
for α ≤ 1
max{0, −α } and n is a sufficiently large integer.
Proof. In view of 3.11 and 3.12 , we can choose a sufficiently large integer n with
m−1 ! β
α
m 1
a
m m
≤
1
n
m−1 i!
i 0
β i ai
ν
≤
n
α i 1
m
1 ! β
α
m 1
a
m m
,
3.14
where the first inequality holds true for all sufficiently large m, and the second one holds true
for all m ∈ N0 .
If we define ρ0 n, then Theorem 3.1 implies that there exists a solution yn : I∞ → C
of the Kummer’s equation such that the inequality given for |y x − yn x | holds true for any
x ∈ In .
4. An Example
We fix α
1, β
10/3, ε > 0, and 0 < ρ < 1. And we define
b0
0,
bm
ε 1
·
s m2
4.1
Journal of Inequalities and Applications
for all m ∈ N, where we set s
9
5/3 2 − ρ / 1 − ρ . We further define
∞
y x
bm xm
4.2
m 0
for any x ∈ Iρ .
Then, we set am
m
β m
1 bm
10 ε
· ,
3 s
a0
am
1
− m
α bm , that is,
4m2 − 6m − 3
3m2 m 1
1
ε 5 ε
≤ ·
s 3 s
4.3
for every m ∈ N. Obviously, all am s are positive, and the sequence {am } is strictly monotone
decreasing, from the 4th term on, to ε/s. More precisely, a0 > a1 < a2 < a3 < a4 > a5 >
a6 > · · · .
Since
a0
10 ε 1 ε
· > ·
3 s 6 s
41 ε
·
36 s
a1
a3 ,
4.4
we get
∞
am xm
a0
a1 x
a2 x2
a3 x3
≥ a0
a1 x
a2 x2
a3 x3
a4 x4
a5 x5
a6 x6
a7 x7
···
m 0
4.5
≥ a0 − a1 − a3
73 ε
·
36 s
for each x ∈ Iρ and
∞
∞
|am xm | ≤
m 0
am ρm ≤
m 0
∞
10
3
m
5 m
ρ
3
1
ε
s
ε
4.6
for all x ∈ Iρ . Hence, we obtain
∞
|am xm | ≤ K
m 0
for any x ∈ Iρ , where K
∞
am xm
m 0
60/73 · 2 − ρ / 1 − ρ , implying that y ∈ CK .
4.7
10
Journal of Inequalities and Applications
We will now show that {am } satisfies condition 3.1 . For any m ∈ N, we have
m−1 i!
i 0
β i ai
α i 1
a0
i 1
≤
m
1! β
α
a
m m
m 1
m−1
≥
10 · 13 · 16 · · · 3i
i 1 3i
m−1
10
3
i 1
7
10 · 13 · 16 · · · 3i
i 1 3i
10 · 13 · 16 · · · 3m
3m
7
·
ai
7
·
5 ε
,
3 s
4.8
1 ε
· ,
6 s
since limm → ∞ am ε/s.
It follows from 4.8 that
m−1 i!
i 0
β i ai
1
≤ 10
3
α i 1
10
≤ 10
≤ 10
m−1
i 1
10 · 13 · 16 · · · 3i
i 1 3i
1
3
10 · 13 · · · 3m
3m
1
3
10 · 13 · 16 · · · 3m
3m
i 1
7
7
5π 2 − 12
3
m
1! β
α
a
m m
m 1
3m−i
10 · · · 3m
3i
m−1
1
i
i 1
1
10
5π 2 − 12 10 · 13 · 16 · · · 3m
·
3
3m
≤
1 ε
6 s
·
m−1
10 · 13 · 16 · · · 3m
3m
7
7
1
2
·
1
ζ 2 −1
6
7
·
7
·
1
i
1 ε
·
1 6 s
1 ε
6 s
4.9
ε
s
1 ε
·
6 s
.
We know that the inequality 4.9 is also true for m 0.
On the other hand, in view of Remark 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 {am } satisfies condition 3.1 with ν
5π 2 − 12 μ/3.
Finally, it follows from 4.6 that
xy x
β − x y x − αy x
∞
am xm ≤
m 0
for all x ∈ Iρ0 with ρ0
min{ρ, 1/μ}.
∞
|am xm | ≤ ε
m 0
4.10
Journal of Inequalities and Applications
11
According to Theorem 3.1, there exists a solution yh : I∞ → C of the Kummer’s
equation 2.1 such that
y x − yh x
≤
100π 2 − 240 2 − ρ
·
ε
73
1−ρ
4.11
for all x ∈ 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 Government No. 2009-0071206 .
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