doi:10.1155/2010/594783
Research Article
Error Bounds for Asymptotic Solutions of
Second-Order Linear Difference Equations II:
The First Case
L. H. Cao
1, 2and J. M. Zhang
31Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong 2Department of Mathematics, Shenzhen University, Guangdong 518060, China
3Department of Mathematics, Tsinghua University, Beijin 100084, China
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
Copyrightq2010 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 equationyn2 npanyn1 nqbnyn 0,wherepandqare integers,anandbnhave asymptotic expansions of the forman∼∞s0as/ns,bn∼
∞
s0bs/ns, for large values ofn,
a0/0,andb0/0.
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 1first deal with this problem. More than eighty years later, this problem was picked up again by Wong and Li2,3. This time two papers on asymptotic solutions to the following difference equations:
yn2 anyn1 bnyn 0 1.1
yn2 npanyn1 nqbnyn 0 1.2
were published, respectively, where coefficientsanandbnhave asymptotic properties
an∼∞
s0 as
ns, bn∼
∞
s0 bs
ns, 1.3
Unlike the method used by Olver4to 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 ask < 0 wherek 2p−q. The whole proof of the result is too long to understand, so we divide the estimations into two parts, part Ithis paperand part IIthe next paper, which correspond to the different two cases of3, respectively.
In the rest of this section, we introduce the main results of3 in the case thatk 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 caseis dedicated to the case when
k <0.
1.1. The Result in [
3
] When
k
1
Whenk 1, from3we know that1.2has two linearly independent solutiony1nand y2n
y1n n−2!q−pρn1nα1 ∞
s0 c1
s
ns , 1.4
ρ1− b0 a0
, α1
b0 a2 0
−a1 a0
b1 b0 −
pq, c1
0 /0, 1.5
y2n n−2!pρ2nnα2 ∞
s0 c2
s
ns , 1.6
ρ2−a0, α2 a1
0
a1− b0 a0
, c2
0 /0, 1.7
1.2. The Result in [
3
] When
k >
1
Whenk >1, from3we know that1.2has two linearly independent solutionsy1nand y2n
y1n n−2!q−pρ1nnα1 ∞
s0 c1
s
ns , 1.8
ρ1 −ba0 0
, α1 bb1
0 − a1 a0 −
pq, c1
0 /0, 1.9
y2n n−2!pρ2nnα2 ∞
s0 c2
s
ns , 1.10
ρ2−a0, α2 aa1 0
, c2
0 /0. 1.11
In the following sections, we will discuss in detail the error bounds of the proceeding asymptotic solutions of1.2. 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 ∞
jn
Kn, jRj−jpφjhj1−jqψjhj .
2.1
Lemma 2.1. LetKn, j,φj,ψj,Rjbe real or complex functions of integer variablesn, j;pand
qare integers. If there exist nonnegative constantsn1,θ,ς,N,s,t,β,CK,CR,Cφ,Cψ,Cβ,Cαwhich
satisfy
−θp−s−β−3
2, −θq−t−2β− 3
2, 2.2
and whenj nn1,
Kn, jCKPn
pjj−θ, R
jCRp
jj−ςN−1,
φ
jCφj−s, ψjCψj−t,
Pj pj
Cβj−2β,
Pj1
pj
2Cαρ1j−β,
wherePnandpnare positive functions of integer variablen. Letn0,n2be integers defined by
n2
1
ς
2CK
2CαCφρ1sup
j
11
j
−ςN−θ−3/2
CψCβ
−θ
,
n0max{n1, n2},
2.4
then2.1has a solutionhn, which satisfies
|hn| 2CRCKPnn−ςN−θ−1 ςNθ−2CK
2CαCφρ1supj
11/j−ςN−θ−3/2CψCβ
,
2.5
forNn0.
Proof. Set
h0n 0,
hs1n ∞
jn
Kn, jRj−jpφjh
sj1−jqψjhsj ,
s0,1,2, . . . ,
2.6
then
|h1n| ∞
jn
Kn, jRjCKCRPn
∞
jn
j−ςN−θ−1
2CRCK
ςNθPnn−ςN−θ.
2.7
The inequality∞jnj−p2/p−1n−p−1, np−1>0, is used here. Assuming that
|hsn−hs−1n| ςN2CRCKθλs−1Pnn−ςN−θ−1, 2.8
where
λ ςN2CKθ
2CαCφρ1sup
j
11
j
−ςN−θ−3/2
CψCβ
then
|hs1n−hsn|
∞
jn
Kn, jjpφjh
sj1hs−1j1jqψjhsj−hs−1j
∞
jn
2CRCK2Pn
ςNθpjj−θ
jp−sC
φλs−1Pj1j1−ςN−θ−1
jq−tC
ψλs−1Pjj−ςN−θ−1
2CRCK
ςNθλsPnn−ςN−θ−1.
2.10
By induction, the inequality holds for any integers. Hence the series
∞
s0
{hs1n−hsn}, 2.11
whenλ <1, that is,Nn0max{n1, n2}, is uniformly convergent innwhere
n2
1
ς
2CK
2CαCφρ1sup
j
11
j
−ςN−θ−3/2
CψCβ
−θ
. 2.12
And its sum
hn ∞
s0
{hs1n−hsn} 2.13
satisfies
|hn|∞
s0
|hs1n−hsn| ςN2CRCKθPnn−ςN−θ−1
∞
s0 λs
2CRCKPnn−ςN−θ−1 ςNθ−2CK
2CαCφρ1supj
11/j−ςN−θ−3/2CψCβ
. 2.14
2.2. The Bound Estimate of a Solution to
a Nonlinear First-Order Difference Equation
Lemma 2.2. If the functionfnsatisfies
fn 1 A
n2 f1n, 2.15
wheren3|f
1n|B(AandBare constants), whennis large enough, then the following first-order
difference equation
xnxn1 fn,
x∞ 1 2.16
has a solutionxnsuch thatsupn{n2|xn−1|}is bounded by a constantC
x, whennis big enough.
Proof. Obviously from the conditions of this lemma, we know that infinite products
∞
k0fn2kand
∞
k0fn2k1are convergent.
xn
∞
k0fn2k
∞
k0fn2k1
. 2.17
is a solution of2.16with the infinite condition. Letgn, k fn2k/fn2k1−1; then whennis large enough,
gn, k4|A|4B n2k3,
n2|xn−1|n2
∞
k0fn2k
∞
k0fn2k1
−1
n2∞
k0
1gn, k−1
4|A|4BCx.
2.18
3. Error Bounds in the Case When
k
1
Before giving the estimations of error bounds of solutions to1.2, we rewriteyinas
with order linear difference equations
εNin2 npanεNin1 nqbnεNin RNin, i1,2, 3.3
3.1. The Error Bound for the Asymptotic Expansion of
y
1n
Now we firstly estimate the error bound of the asymptotic expansion ofy1nin the case k1. Let
It can be easily verified that
z1n n−2!q−pρn1nα1
are two linear independent solutions of the comparative difference equation
From the definition, we know that the two-term approximation ofxnis
Equation 3.8 is a second-order linear difference equation with two known linear independent solutions. Its coefficients are quite similar to those in3.3. This reminds us to rewrite3.3in the form similar to3.8.
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.
From10, any solution of the “summary equation”
Firstly we consider the denominator inKn, j. We get from3.8
Set the Wronskian of the two solutions of the comparative difference equation as
3.2. The Error Bound for the Asymptotic Expansion of
y
2(
n
)
Now we estimate the error bound of the asymptotic expansion of the linear independent solutiony2nto the original difference equation ask1. Let
ε2
Nn y1nδNn. 3.33
From3.3, we have
y1n2δNn2 npany1n1δNn1 nqbny1nδNn RN2n. 3.34
Fory1nbeing a solution of1.2, let
ΔNn δNn1−δNn; 3.35
thenΔNnsatisfies the first-order linear difference equation
y1n2ΔNn1−nqbny1nΔNn RN2n. 3.36
The solution of3.36is
ΔNn −
∞
in
Xn
Xi1
R2
Ni
y1i2
, 3.37
where Xn Xmnj−1mjqbjy1j/y1j 2, Xm is a constant, andmis an integer
which is large enough such that wheninm,
y1i i−2!p−1ρi1iReα11ε 1 1 i
1
2i−2! p−1ρi
1iReα1. 3.38
The two-term approximation ofjqbjy1j/y1j2is
jqbjy
1
j y1
j2
b0j ρ2
1
1α2−α1
j σ
j, 3.39
whereσjis the reminder andσ0supj{j2|σj|}is a constant. From Lemma 3 of5, we obtain
|Xm|b0 ρ2 1
n−m
n−1!
m−1!exp−k1n
Reα2−α1
|Xn||Xm|b0 ρ2 1
n−m
n−1!
m−1!expk1n
Reα2−α1,
LetM0 Ii0−1mi−1!|ρ2/ρ1|
iiReα2−α1−N; then
n−1
im
i−1!ρ2
ρ1
iiReα2−α1−NM
0nn−1! ρ2
ρ1
nnReα2−α1−N
2n−2!ρ2
ρ1
nnReα2−α1−N1,
3.47
where limn→ ∞n−2!|ρ2/ρ1|
nnReα2−α1−N2∞.Hence
|δNn|2μn−2!ρρ2 1
nnReα2−α1−N2. 3.48
From3.33, we obtain
ε2
Nny1nδNn
≤2μsup nm
n−2!1−pρ1−nn−Reα1y 1n
×n−2!pρ2
nnReα2−N2.
3.49
Thus we complete the estimate of error bounds to asymptotic expansions of solutions of1.2
ask1.
4. Error Bounds in Case When
k >
1
Here we also rewriteyinas
yin LNin εNin, i1,2, 4.1
with
L1
Nn n−2!q−pρn1nα1
N−1
s0 c1
s
ns,
L2
Nn n−2!pρ2nnα2
N−1
s0 c2
s
ns ,
4.2
andεNin,i 1,2, are error terms. ThenεNin,i 1,2, satisfy the inhomogeneous second-order linear difference equations
where
4.1. The Error Bound for the Asymptotic Expansion of
y
1n
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.
is a constant. And the function
such that
are two linear independent solutions of the difference equation
zn2 np
This difference equation4.13can be regarded as the comparative equation of4.3. Rewriting4.3in the form similar to the comparative difference equation4.13, we get
ε1 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 equation4.13.
From10, any solution of the “summary equation”
Similar toSection 3.1, we have
we have fromLemma 2.1that
4.2. The Error Bound for the Asymptotic Expansion of
y
2n
Letε2
From3.3, we have
y1n2δNn2 npany1n1δNn1 nqbny1nδNn RN2n. 4.23
Using the method employed inSection 3.2, it is not difficult to obtain
ε2
Nny1nδNn
2μsup nm
n−2!kρ1−nn−Reα1y1nρ2nnReα2−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−yn1n1
2yn0. 4.25
It is a special case of the equation
n1fnα1x−nαxfnαx fnα−1x 0, 4.26
α1, x1, which is satisfied by Tricomi-Carlitz polynomials. By calculation, the constant
CKin3.30is1728/5e2π2. So4.25has a solution
y1n n−1
2!n21ε1n, 4.27
forn≥3 with the error termε1nsatisfing
|ε1n|
864 25e
2π2
n−1. 4.28
Acknowledgments
References
1 G. D. Birkhoff, “General theory of linear difference equations,” Transactions of the American Mathematical Society, vol. 12, no. 2, pp. 243–284, 1911.
2 R. Wong and H. Li, “Asymptotic expansions for second-order linear difference equations,”Journal of Computational and Applied Mathematics, vol. 41, no. 1-2, pp. 65–94, 1992.
3 R. Wong and H. Li, “Asymptotic expansions for second-order linear difference equations. II,”Studies in Applied Mathematics, vol. 87, no. 4, pp. 289–324, 1992.
4 F. W. J. Olver, Asymptotics and Special Functions, Computer Science and Applied Mathematics, Academic Press, New York, NY, USA, 1974.
5 J. M. Zhang, X. C. Li, and C. K. Qu, “Error bounds for asymptotic solutions of second-order linear difference equations,”Journal of Computational and Applied Mathematics, vol. 71, no. 2, pp. 191–212, 1996.
6 R. Spigler and M. Vianello, “Liouville-Green approximations for a class of linear oscillatory difference equations of the second order,”Journal of Computational and Applied Mathematics, vol. 41, no. 1-2, pp. 105–116, 1992.
7 R. Spigler and M. Vianello, “WKBJ-type approximation for finite moments perturbations of the differential equationy
0 and the analogous difference equation,”Journal of Mathematical Analysis and Applications, vol. 169, no. 2, pp. 437–452, 1992.
8 R. Spigler and M. Vianello, “Discrete and continuous Liouville-Green-Olver approximations: a unified treatment via Volterra-Stieltjes integral equations,”SIAM Journal on Mathematical Analysis, vol. 25, no. 2, pp. 720–732, 1994.
9 R. Spigler and M. Vianello, “A survey on the Liouville-GreenWKB approximation for linear difference equations of the second order,” inAdvances in Difference Equations (Veszpr´em, 1995), N. Elaydi, I. Gy ¨ori, and G. Ladas, Eds., pp. 567–577, Gordon and Breach, Amsterdam, The Netherlands, 1997.