Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 594783, 19 pages
doi:10.1155/2010/594783
Research Article
Error Bounds for Asymptotic Solutions of
Second-Order Linear Difference Equations II:
The First Case
L. H. Cao1, 2 and J. M. Zhang3
1
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
Department of Mathematics, Shenzhen University, Guangdong 518060, China
3
Department of Mathematics, Tsinghua University, Beijin 100084, China
2
Correspondence should be addressed to J. M. Zhang, jzhang@math.tsinghua.edu.cn
Received 13 July 2010; Accepted 27 October 2010
Academic Editor: Rigoberto Medina
Copyright q 2010 L. H. Cao and J. M. Zhang. 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 discuss in detail the error bounds for asymptotic solutions of second-order linear difference
equation yn 2 np anyn 1 nq bnyn
p and
q are integers, an and bn have
0, where
∞
s
s
asymptotic expansions of the form an ∼ ∞
s0 as /n , bn ∼
s0 bs /n , for large values of n,
0, and b0 /
0.
a0 /
1. Introduction
Asymptotic expansion of solutions to second-order linear difference equations is an old
subject. The earliest work as we know can go back to 1911 when Birkhoff 1 first deal with
this problem. More than eighty years later, this problem was picked up again by Wong and
Li 2, 3. This time two papers on asymptotic solutions to the following difference equations:
yn 2 anyn 1 bnyn 0
p
q
yn 2 n anyn 1 n bnyn 0
1.1
1.2
were published, respectively, where coefficients an and bn have asymptotic properties
an ∼
∞
as
s0
,
ns
for large values of n, a0 /
0, b0 /
0, and p, q ∈ Z.
bn ∼
∞
bs
,
s
n
s0
1.3
2
Advances in Difference Equations
Unlike the method used by Olver 4 to treat asymptotic solutions of second-order
linear differential equations, the method used in Wong and Li’s papers cannot give us way
to obtain error bounds of these asymptotic solutions. Only order estimations were given
in their papers. The estimations of error bounds for these asymptotic solutions to 1.1
were given in 5 by Zhang et al. But the problem of obtaining error bounds for these
asymptotic solutions to 1.2 is still open. The purpose of this and the next paper Error
bounds for asymptotic solutions of second-order linear difference equations II: the second
case is to estimate error bounds for solutions to 1.2. The idea used in this paper is
similar to that of Olver to obtain error bounds to the Liouville-Green WKB asymptotic
expansion of solutions to second-order differential equations. It should be pointed out
that similar method appeared in some early papers, such as Spigler and Vianello’s papers
6–9.
In Wong and Li’s second paper 3, two different cases were given according to
different values of parameters. The first case is devoted to the situation when k > 0, and
in the second case as k < 0 where k 2p − q. The whole proof of the result is too long to
understand, so we divide the estimations into two parts, part I this paper and part II the
next paper, which correspond to the different two cases of 3, respectively.
In the rest of this section, we introduce the main results of 3 in the case that k is
positive. In the next section, we give two lemmas on estimations of bounds for solutions
to a special summation equation and a first order nonlinear difference equation which will
be often used later. Section 3 is devoted to the case when k 1. And in Section 4, we
discuss the case when k > 1. The next paper Error bounds for asymptotic solutions of
second-order linear difference equations II: the second case is dedicated to the case when
k < 0.
1.1. The Result in [3] When k 1
When k 1, from 3 we know that 1.2 has two linearly independent solution y1 n and
y2 n
y1 n n − 2!q−p ρ1n nα1
ρ1 −
b0
,
a0
α1 for n ≥ 2.
s0
ns
1.4
,
b0 a1 b1
−
− p q,
a20 a0 b0
y2 n n − 2!p ρ2n nα2
ρ2 −a0 ,
1
∞
cs
α2 1
c0 /
0,
∞ c2
s
,
s
n
s0
1
b0
a1 −
,
a0
a0
1.5
1.6
2
c0 /
0,
1.7
Advances in Difference Equations
3
1.2. The Result in [3] When k > 1
When k > 1, from 3 we know that 1.2 has two linearly independent solutions y1 n and
y2 n
y1 n n − 2!q−p ρ1n nα1
ρ1 −
b0
,
a0
α1 ∞ 1
cs
s0
ns
b1 a1
−
− p q,
b0 a0
y2 n n − 2!p ρ2n nα2
ρ2 −a0 ,
α2 a1
,
a0
1
c0 /
0,
∞ 2
cs
s0
1.8
,
ns
,
2
c0 /
0.
1.9
1.10
1.11
In the following sections, we will discuss in detail the error bounds of the proceeding
asymptotic solutions of 1.2. Before discussing the error bounds, we consider some lemmas.
2. Lemmas
2.1. The Bounds for Solutions to the Summation Equation
We consider firstly a bound of a special solution for the “summary equation”
hn ∞
K n, j R j − j p φ j h j 1 − j q ψ j h j .
jn
2.1
Lemma 2.1. Let Kn, j, φj, ψj, Rj be real or complex functions of integer variables n, j; p and
q are integers. If there exist nonnegative constants n1 , θ, ς, N, s, t, β, CK , CR , Cφ , Cψ , Cβ , Cα which
satisfy
3
−θ p − s − β − ,
2
3
− θ q − t − 2β − ,
2
2.2
and when j n n1 ,
P n
R j CR p j j −ςN−1 ,
K n, j CK j −θ ,
p j
ψ j Cψ j −t ,
φ j Cφ j −s ,
P j 1
P j
−2β
2Cα ρ1 j −β ,
Cβ j ,
p j
p j
2.3
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Advances in Difference Equations
where P n and pn are positive functions of integer variable n. Let n0 , n2 be integers defined by
1
1 −ςN−θ−3/2
Cψ Cβ − θ ,
n2 2CK 2Cα Cφ ρ1 sup 1 ς
j
j
2.4
n0 max{n1 , n2 },
then 2.1 has a solution hn, which satisfies
2CR CK P nn−ςN−θ−1
|hn| ςN θ − 2CK 2Cα Cφ ρ1 supj 1 1/j
−ςN−θ−3/2
,
Cψ Cβ
2.5
for N n0 .
Proof. Set
h0 n 0,
hs1 n ∞
K n, j R j − j p φ j hs j 1 − j q ψ j hs j ,
jn
2.6
s 0, 1, 2, . . . ,
then
|h1 n| ∞
∞
K n, j R j CK CR P n
j −ςN−θ−1
jn
The inequality
∞
jn
jn
2.7
2CR CK
P nn−ςN−θ .
ςN θ
j −p 2/p − 1 n−p−1 , n p − 1 > 0, is used here. Assuming that
|hs n − hs−1 n| 2CR CK s−1
λ P nn−ςN−θ−1 ,
ςN θ
2.8
where
2CK
λ
ςN θ
1
2Cα Cφ ρ1 sup 1 j
j
−ςN−θ−3/2
Cψ Cβ ;
2.9
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5
then
|hs1 n − hs n| ∞
K n, j j p φ j hs j 1 hs−1 j 1 j q ψ j hs j −hs−1 j
jn
∞ 2C C2 P n
−ςN−θ−1
R
K
j −θ j p−s Cφ λs−1 P j 1 j 1
jn ςN θp j
j q−t Cψ λs−1 P j j −ςN−θ−1
2CR CK s
λ P nn−ςN−θ−1 .
ςN θ
2.10
By induction, the inequality holds for any integer s. Hence the series
∞
{hs1 n − hs n},
2.11
s0
when λ < 1, that is, N n0 max{n1 , n2 }, is uniformly convergent in n where
1
1 −ςN−θ−3/2
n2 2CK 2Cα Cφ ρ1 sup 1 Cψ Cβ − θ .
ς
j
j
2.12
And its sum
hn ∞
{hs1 n − hs n}
2.13
s0
satisfies
|hn| ∞
∞
2CR CK
P nn−ςN−θ−1 λs
|hs1 n − hs n| ςN θ
s0
s0
2CR CK P nn−ςN−θ−1
−ςN−θ−3/2
ςN θ − 2CK 2Cα Cφ ρ1 supj 1 1/j
Cψ Cβ
.
2.14
So we get the bound of any solution for the “summary equation” 2.1. Next we
consider a nonlinear first-order difference equation.
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Advances in Difference Equations
2.2. The Bound Estimate of a Solution to
a Nonlinear First-Order Difference Equation
Lemma 2.2. If the function fn satisfies
A
f1 n,
n2
fn 1 2.15
where n3 |f1 n| B (A and B are constants), when n is large enough, then the following first-order
difference equation
xnxn 1 fn,
x∞ 1
2.16
has a solution xn such that supn {n2 |xn − 1|} is bounded by a constant Cx , when n is big enough.
Proof. Obviously from the conditions of this lemma, we know that infinite products
∞
∞
k0 fn 2k and
k0 fn 2k 1 are convergent.
∞
fn 2k
.
xn ∞ k0
k0 fn 2k 1
2.17
is a solution of 2.16 with the infinite condition. Let gn, k fn 2k/fn 2k 1 − 1;
then when n is large enough,
gn, k 4|A| 4B
,
n 2k3
∞
fn 2k
−1
n2 |xn − 1| n2 ∞ k0
fn
2k 1
k0
n2
∞
2.18
1 gn, k − 1
k0
4|A| 4B Cx .
3. Error Bounds in the Case When k 1
Before giving the estimations of error bounds of solutions to 1.2, we rewrite yi n as
i
i
yi n LN n εN n,
i 1, 2,
3.1
Advances in Difference Equations
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with
1
LN n n − 2!q−p ρ1n nα1
2
LN n n −
i
N−1
cs1
,
ns
s0
3.2
N−1
cs2
,
2!p ρ2n nα2
ns
s0
i
and εN n, i 1, 2, being error terms. Then εN n, i 1, 2, satisfy inhomogeneous secondorder linear difference equations
i
i
i
i
εN n 2 np anεN n 1 nq bnεN n RN n,
i 1, 2,
3.3
where
i
i
i
i
RN n − LN n 2 np anLN n 1 nq bnLN n ,
i 1, 2.
3.4
We know from 3 that
N
CR1 sup n
n
1
RN n
n!q−p ρ1n nα1
,
N1
CR2 sup n
n
2
RN n
n!p ρ2n nα2
.
3.5
3.1. The Error Bound for the Asymptotic Expansion of y1 n
Now we firstly estimate the error bound of the asymptotic expansion of y1 n in the case
k 1. Let
1 − 1/nρ22 1 2/nα2 − n−2 ρ12 1 2/nα1
,
xn − ρ2 1 − 1/n1 1/nα2 − ρ1 n−1 1 1/nα1 a0 a1 /n
n1 − 1/np ρ22 1 2/nα2 a0 a1 /nρ2 1 1/nα2 xn 1
b1
ln −
− b0 − .
xnxn 1
n
3.6
It can be easily verified that
z1 n n − 2!q−p ρ1n nα1
z2 n n −
2!p ρ2n nα2
∞
xk,
kn
∞
xk
3.7
kn
are two linear independent solutions of the comparative difference equation
zn 2 n
p
a1
b1
q
zn 1 n b0 ln zn 0.
a0 n
n
3.8
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Advances in Difference Equations
From the definition, we know that the two-term approximation of xnis
a0 a1 − b0 /a0 − pa20 − a1 − pa0 a0 b0 1
ωn,
xn 1 n
a20
3.9
where ωn is the reminder and the coefficient of 1/n is zero. So Cx supn {n2 |xn − 1|} is
a constant. And ln satisfies Cl supn {n2 |ln|} being a constant; here we have made use of
the definitions of αi , ρi in 1.5, 1.7, and 2p − q 1.
Equation 3.8 is a second-order linear difference equation with two known linear
independent solutions. Its coefficients are quite similar to those in 3.3. This reminds us to
rewrite 3.3 in the form similar to 3.8.
According to the coefficients in 3.8, we rewrite 3.3 as
a1 1
b1
1
q
εN n 1 n b0 ln εN n
2 n a0 n
n
a1 1
b1
1
1
p
q
− ln εN n,
ε n 1 − n bn − b0 −
RN n − n an − a0 −
n N
n
1
εN n
p
3.10
where an and bn are such that
a1
2
,
Ca sup j a j − a0 −
j
jn
b1
2
Cb sup j b j − b0 −
−l j
j
jn
3.11
are finite. Equation 3.10 is a inhomogeneous second-order linear difference equation; its
solution takes the form of a particular solution added to an arbitrary linear combination of
solutions to the associated homogeneous linear difference equation3.8.
From 10, any solution of the “summary equation”
1
εN n ∞
1 a1 1 K n, j RN j − j p a j − a0 −
εN j 1
j
jn
1 b1
q
− l j εN j
−j b j − b0 −
j
3.12
is a solution of 3.10, where
z1 j 1 z2 n − z1 nz2 j 1
K n, j .
z1 j 2 z2 j 1 − z1 j 1 z2 j 2
Now we estimate the bound of the function Kn, j.
3.13
Advances in Difference Equations
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Firstly we consider the denominator in Kn, j. We get from3.8
z1 n 2z2 n 1 − z1 n 1z2 n 2
b1
ln z1 nz2 n 1 − z1 n 1z2 n 0.
nq b0 n
3.14
Set the Wronskian of the two solutions of the comparative difference equation as
Wn z1 n 1z2 n − z1 nz2 n 1;
3.15
b1
ln Wn.
Wn 1 nq b0 n
3.16
we have
From 3.16, we have
Wn 1 W2n!q b0n−1
n 1
k2
b1 1 lk
.
b0 k
b0
3.17
From Lemma 3 of 5, we obtain
exp−k1 n 1Reb1 /b0 n km
1
b1 1 lk
b0 k
b0
3.18
expk1 n 1Reb1 /b0 ,
where
1
1
1
π2
σ0 ,
ln
m
m 6m2 60m4
6
b1 1 lk
b1 1
−
σ0 sup k2 ln 1 m < k < n;
b0 k
b0
b0 k
k
k1 b1
b0
3.19
3.20
m is an integer which is large enough such that 1 b1 /b0 1/k lk/b0 > 0, when k m.
Let C∗ | m−1
k2 1 b1 /b0 1/k lk/b0 |, for the property of lk, we know that
C∗ is a constant. Then we obtain from 3.18
|Wn 1| |W2|n!q b0n−1 C∗ exp−k1 n 1Reb1 /b0 .
3.21
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Advances in Difference Equations
Now considering the numerator in Kn, j, we get
z1 j 1 z2 n − z1 nz2 j 1
∞
∞
p−1
j −1 !
xk xk
n − 2!p−1
kj1
kn
3.22
j1 α
α
j1 × ρ1 ρ2n j 1 1 nα2 n − 2! − ρ1n ρ2 j 1 2 nα1 j − 1 ! .
Here we have made use of q − p p − 1.
From Lemma 2 of 5, we have
∞
xk
kj1
∞
xk exp
kn
2π 2
Cx ,
3
3.23
where Cx supn {n2 |xn − 1|} is a constant. For the bound of Kn, j, we set
n − 2!q−p ρ1n nα1 n, j ,
K n, j K
q−p j α
ρ1 j 1
j!
3.24
n, j |I| |II|,
K
3.25
then
where
|I| q−p j α
j!
ρ1 j 1
n − 2!q−p ρ1n nα1
p−1
exp 2π 2 /3 Cx j − 1 !
n − 2!p−1
Reb1 /b0 q j
|W2| j! b0 C∗ exp−k1 j 1
j1 × ρ1n ρ2
|II| α
j 1 2 nα1 j − 1 !
q−p j α
j!
ρ1 j 1
n − 2!q−p ρ1n nα1
p−1
exp 2π /3 Cx j − 1 !
n − 2!p−1
Reb1 /b0 q j
|W2| j! b0 C∗ exp−k1 j 1
α
j1
× ρ1 ρ2n j 1 1 nα2 n − 2! .
2
3.26
Advances in Difference Equations
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By simple calculations, we get
|I| exp 2π 2 /3 Cx k1
1 α2 −Reb1 /b0 −1
ρ
sup
j .
1
2
j
|W2|C∗
jn
3.27
Here we have made use of 1.5 and 1.7.
Since |n − 2!/j − 1!ρ2 /ρ1 n−j n/jα1 −α2 | 1, we have
exp 2π 2 /3 Cx k1
1 α1 −Reb1 /b0 −1
ρ
sup
j .
1
|II| 1
j
|W2|C∗
jn
3.28
Here we also have made use of 1.5 and 1.7.
Let
exp 2π 2 /3 Cx k1
CK |W2|C∗
1
ρ2 sup 1 j
jn
α2 −Reb1 /b0 1 α1 −Reb1 /b0 ,
ρ1 sup 1 j
jn
3.29
we have from 3.24 the bound of Kn, j
n − 2!p−1 ρ1n nα1 −1
K n, j CK
j .
p−1 j α
j!
ρ1 j 1
3.30
1
For the bound of εN n, set P n n − 2!q−p ρ1n nα1 , pn n!q−p ρ1n nα1 , θ 1, Rj Ca , Cψ Cb , CR CR1 , s t 2, Cβ supjn 1 − 1/j1−p , β p − 1, Cα supjn 1 1/jα1 , ς 1; we have from Lemma 2.1 that
1
RN j, Cφ
1
εN n ≤ n − 2!q−p ρ1n nα1 n−N−1
×
2CR1 CK
,
−N−5/2
N − 2Ck 2Cα Ca ρ1 supj≥n 1 1/j
Cb Cβ
3.31
when
2CK
λ
N
2Cα Ca ρ1 sup 1 1/j
jn
−N−5/2
Cb Cβ
< 1,
3.32
that is, N ≥ n0 2CK 2Cα Ca |ρ1 |supjn 1 1/j−N−5/2 Cb Cβ − 1 and j ≥ n ≥ N ≥ n0 .
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Advances in Difference Equations
3.2. The Error Bound for the Asymptotic Expansion of y2 (n)
Now we estimate the error bound of the asymptotic expansion of the linear independent
solution y2 n to the original difference equation as k 1. Let
2
εN n y1 nδN n.
3.33
From 3.3, we have
2
y1 n 2δN n 2 np any1 n 1δN n 1 nq bny1 nδN n RN n.
3.34
For y1 n being a solution of 1.2, let
ΔN n δN n 1 − δN n;
3.35
then ΔN n satisfies the first-order linear difference equation
2
y1 n 2ΔN n 1 − nq bny1 nΔN n RN n.
3.36
The solution of 3.36 is
ΔN n −
2
∞
Xn RN i
,
Xi 1 y1 i 2
in
3.37
q
where Xn Xm n−1
jm j bjy1 j/y1 j 2, Xm is a constant, and m is an integer
which is large enough such that when i n m,
1
y1 i i − 2!p−1 ρ1i iRe α1 1 ε1 i 1
i − 2!p−1 ρ1i iRe α1 .
2
3.38
The two-term approximation of j q bjy1 j/y1 j 2 is
j q b j y1 j
b0 j
α2 − α1
σ j ,
2 1
j
y1 j 2
ρ1
3.39
where σj is the reminder and σ0 supj {j 2 |σj|} is a constant.
From Lemma 3 of 5, we obtain
|Xm|
b0
ρ12
n−m
n − 1!
exp−k1 nReα2 −α1 m − 1!
b0
|Xn| |Xm| 2
ρ1
n−m
n − 1!
expk1 nReα2 −α1 ,
m − 1!
3.40
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where
1
1
π2
1
σ0 ,
ln
m
k1 |α2 − α1 |
m 6m2 60m4
6
α2 − α1
α2 − α1
2
σ j −
σ0 sup j ln 1 ,
j
j
j
3.41
are constants.
Substituting 3.38 and 3.40 into 3.37, we get
|ΔN n| ≤
ρ n
2CR2 e2k1
n − 1!nReα2 −α1 2
ρ1
|b0 |
Re α2
∞
i
i 1 Re α1 ×sup
sup
i−N−1
i
1
i
2
i≥n
i≥n
in
ρ n
4CR2 e2k1
n − 1!nReα2 −α1 2
ρ1
|b0 |
Re α2
i
i 1 Re α1 n−N
.
×sup
sup
i1
i2
N
in
in
3.42
Let μ 4CR2 e2k1 /|b0 |supin i/i 1Re α2 supin i 1/i 2Re α1 1/N; then
|ΔN n| μn − 1!
ρ2
ρ1
n
nReα2 −α1 −N.
3.43
From 3.35, we have
δN n δN m n−1
ΔN i n i m,
3.44
im
where δN m is a constant. Let δN m 0; we have
|δN n| For
n−1
im i
n−1
n−1
ρ2 i Reα2 −α1 −N
i
.
|ΔN i| μ i − 1!
ρ1
im
im
3.45
− 1!|ρ2 /ρ1 |i iReα2 −α1 −N , there exists a positive integer I0 such that
i! ρ2 /ρ1
i1
i 1Reα2 −α1 −N
i
i − 1! ρ2 /ρ1 iReα2 −α1 −N
ρ
i 2
ρ1
1 Reα2 −α1 −N
1
1,
i
3.46
when i I0 . Thus the sequence {i − 1!|ρ2 /ρ1 |i iReα2 −α1 −N } is increasing when i I0 m.
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Advances in Difference Equations
Let M0 I0 −1
im i
− 1!|ρ2 /ρ1 |i iReα2 −α1 −N ; then
n−1
ρ
ρ2 i Reα2 −α1 −N
i
M0 nn − 1! 2
i − 1!
ρ1
ρ1
im
n
ρ
2n − 2! 2
ρ1
n
nReα2 −α1 −N
3.47
nReα2 −α1 −N1 ,
where limn → ∞ n − 2!|ρ2 /ρ1 |n nReα2 −α1 −N2 ∞. Hence
|δN n| 2μn − 2!
ρ2
ρ1
n
nReα2 −α1 −N2 .
3.48
From 3.33, we obtain
2
εN n y1 nδN n
≤ 2μ sup n − 2!
1−p
nm
−n
ρ1
− Re α1
n
y1 n
3.49
n
×n − 2!p ρ2 nRe α2 −N2 .
Thus we complete the estimate of error bounds to asymptotic expansions of solutions of 1.2
as k 1.
4. Error Bounds in Case When k > 1
Here we also rewrite yi n as
i
i
yi n LN n εN n,
i 1, 2,
4.1
with
1
LN n n − 2!q−p ρ1n nα1
2
LN n n −
i
N−1
cs1
,
ns
s0
4.2
N−1
cs2
,
2!p ρ2n nα2
ns
s0
i
and εN n, i 1,2, are error terms. Then εN n, i 1,2, satisfy the inhomogeneous secondorder linear difference equations
i
i
i
i
εN n 2 np anεN n 1 nq bnεN n RN n,
i 1, 2,
4.3
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15
where
i
i
i
i
RN n − LN n 2 np anLN n 1 nq bnLN n ,
i 1, 2.
4.4
We know from 3 that
N
CR1 sup n
n
1
RN n
,
n!q−p ρ1n nα1
N1
CR2 sup n
n
2
RN n
n!p ρ2n nα2
.
4.5
4.1. The Error Bound for the Asymptotic Expansion of y1 n
Now let us come to the case when k > 1. This time a difference equation which has two
known linear independent solutions is also constructed for the purpose of comparison for
1.2.
Since
−n−q b−1 n
ρ2 np n 2α2 /n 1α2 − ρ1 nq−p n 2α1 /n 1α1
nα2 /ρ
2 n
p
α2
− 1 n 1 −
nα1 /ρ
1 n
− 1
q−p
n 1
α1
1
A
f1 n,
n2
4.6
where
2b1
1
a1 a1
−pq1 −1
p−q
2
b0
2a0 a0
1 b1 a1
a1 b1
−
−pq
−pq−3 ,
2 b0 a0
a0 b0
A −1 4.7
is a constant and n3 |f1 n| B B is a constant, from Lemma 2.2, we know the difference
equation
xnxn 1 1 A
f1 n,
n2
4.8
with condition x∞ 1 having a solution xn such that
Cx sup n2 |xn − 1|
n
4.9
is a constant. And the function
ln −a0 −
ρ2
n−k bnxn
2 α2
a1
1 −α2
−
1
−
1
n 1 − 1/np 1 1/nα2 ρ2 xn 1
n
n
4.10
16
Advances in Difference Equations
such that
Cl sup n2 |ln|
4.11
n
is a constant. Here we have made use of the definitions of ρ1 , α1 , ρ2 , α2 in 1.9, 1.11 and
q − p p − k.
Obviously functions
z1 n n − 2!q−p ρ1n nα1
∞
xk,
kn
∞
xk
z2 n n − 2!p ρ2n nα2
4.12
kn
are two linear independent solutions of the difference equation
a1
zn 2 np a0 ln zn 1 nq bnzn 0.
n
4.13
This difference equation4.13 can be regarded as the comparative equation of 4.3.
Rewriting 4.3 in the form similar to the comparative difference equation 4.13, we get
a1
1
1
1
εN n 2 np a0 ln εN n 1 nq bnεN n
n
a1
1
1
− ln εN n 1,
RN n − np an − a0 −
n
4.14
where an has the property that Ca supn {n2 |an − a0 − a1 /n − ln|} is a constant. Equation
4.14 is an inhomogeneous second-order linear difference equation; its solution takes the
form of a particular solution added to an arbitrary linear combination of solutions to the
associated homogeneous linear difference equation 4.13.
From 10, any solution of the “summary equation”
1
εN n
∞
1 1 p a1
− l j εN j 1 ,
K n, j RN j − j a j − a0 −
j
jn
4.15
where
z1 j 1 z2 n − z1 nz2 j 1
K n, j ,
z1 j 2 z2 j 1 − z1 j 1 z2 j 2
is a solution of 4.14.
4.16
Advances in Difference Equations
17
Similar to Section 3.1, we have
n, j
K
exp 2π 2 /3 Cx k1 −k
j
|W2|C∗
1 α2 −Reb1 /b0 1 α1 −Reb1 /b0 .
× ρ2 sup 1 ρ1 sup 1 j
j
jn
jn
4.17
Let
CK exp 2π 2 /3 Cx k1
|W2|C∗
1 α2 −Reb1 /b0 1 α1 −Reb1 /b0 ;
× ρ2 sup 1 ρ1 sup 1 j
j
jn
jn
4.18
we get
n − 2!p−k ρ1n nα1
K n, j CK
.
p−k j α
ρ1 j 1
j!
4.19
1
Set P n n − 2!p−k ρ1n nα1 , pn n!p−k ρ1n nα1 , θ k, Rj RN j, Cφ Ca ,
Cψ 0, CR CR1 , s 2, β p − k, Cβ supjn 1 − 1/j−p−k , Cα supjn 1 1/jα1 , ς 1;
we have from Lemma 2.1 that
1
εN n n − 2!p−k ρ1n nα1 n−N−K
2CR1 CK
−N−k−1/2
N k − 4Ck Cα Ca ρ1 supjn 1 1/j
4.20
when
4CK
1 −N−k−1/2
λ
Cα Ca ρ1 sup 1 < 1.
Nk
j
jn
that is,
1 −N−k−1/2
−k ,
N n0 4CK Cα Ca |ρ1 |sup 1 j
jn
4.21
j n N n0 .
4.2. The Error Bound for the Asymptotic Expansion of y2 n
Let
2
εN n y1 nδN n.
4.22
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Advances in Difference Equations
From 3.3, we have
2
y1 n 2δN n 2 np any1 n 1δN n 1 nq bny1 nδN n RN n.
4.23
Using the method employed in Section 3.2, it is not difficult to obtain
2
εN n y1 nδN n
2μ sup n − 2!k ρ1
−n − Re α1
n
nm
y1 n
n
ρ2 nRe α2 −Nk1 .
4.24
Now we completed the estimate of the error bounds for asymptotic solutions to second
order linear difference equations in the first case. For the second case, we leave it to the second
part of this paper: Error Bound for Asymptotic Solutions of Second-order Linear Difference
Equation II: the second case.
In the rest of this paper, we would like to give an example to show how to use the
results of this paper to obtain error bounds of asymptotic solutons to second-order linear
difference equations. Here the difference equation is
yn2 − yn1 1
yn 0.
n2
4.25
It is a special case of the equation
α
α
n 1fn1
x − n αxfnα x fn−1
x 0,
4.26
α 1, x 1, which is satisfied by Tricomi-Carlitz polynomials. By calculation, the constant
2
CK in 3.30 is 1728/5e2π . So 4.25 has a solution
y1 n 1
1 ε1 n,
n − 2!n2
4.27
for n ≥ 3 with the error term ε1 n satisfing
|ε1 n| 864 2π 2 −1
e n .
25
4.28
Acknowledgments
The authors would like to thank Dr. Z. Wang for his helpful discussions and suggestions.
The second author thanks Liu Bie Ju Center for Mathematical Science and Department of
Mathematics of City University of Hong Kong for their hospitality. This work is partially
supported by the National Natural Science Foundation of China Grant no. 10571121 and
Grant no. 10471072 and Natural Science Foundation of Guangdong Province Grant no.
5010509.
Advances in Difference Equations
19
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