Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 743973, 4 pages
http://dx.doi.org/10.1155/2013/743973
Research Article
Developing a Series Solution Method of 𝑞-Difference Equations
Hsuan-Ku Liu
Department of Mathematics and Information Education, National Taipei University of Education, Taiwan
Correspondence should be addressed to Hsuan-Ku Liu; hkliu.nccu@gmail.com
Received 16 November 2012; Accepted 14 April 2013
Academic Editor: Filomena Cianciaruso
Copyright © 2013 Hsuan-Ku Liu. 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.
The series solution is widely applied to differential equations on R but is not found in 𝑞-differential equations. Applying the Taylor
and multiplication rule of two generalized polynomials, we develop a series solution of linear homogeneous 𝑞-difference equations.
As an example, the series solution method is used to find a series solution of the second-order 𝑞-difference equation of Hermite’s
type.
1. Introduction
Several works have been done recently for series solutions on
certain time scales. One of the difficulties for developing a
theory of series solutions for linear homogeneous equations
on time scales is that the formula for multiplication by two
generalized polynomials is not easily found. If the time scale
has constant graininess, Haile and Hall [1] provided an exact
formula for the multiplicity of two generalized polynomials.
Using the obtained results, the series solutions for linear
dynamic equations are proposed on the time scales R and T =
ℎZ (difference equations with step size ℎ). On generalized
time scales, Mozyrska and Pawłuszewicz [2] presented the
formula for the multiplicity of the generalized polynomials
of degree one and degree 𝑛 ∈ N.
Let 0 < 𝑞 < 1 and use the notations
𝑞N = {𝑞𝑛 | 𝑛 ∈ N} ,
𝑞N = 𝑞N ∪ {0} ,
(1)
where N denotes the set of positive integers. Liu [3] presented
a formula for the multiplication of two 𝑞-polynomials. The
obtained results are used to develop a series solution method
of the second-order difference equations on 𝑞N . Precisely, the
second-order 𝑞-difference equation is described as
where 𝑓 and 𝑔 are both 𝑞-analytic functions at 0 in the
interval (𝑐, 𝑑). As an example, the series solution method is
applied to consider the 𝑞-Hermite’s equation of the form
𝑢ΔΔ (𝑡) − 𝑡𝑢Δ (𝑡) + 𝜆𝑢 (𝑡) = 0,
𝑡 ∈ 𝑞N
(3)
with initial condition 𝑢(0) = 𝑎 and 𝑢Δ (0) = 𝑏.
This paper is organized as follows: in Section 2, basic ideas
on 𝑞-calculus are introduced. The series solution method
is developed in Section 3 and is applied to consider the 𝑞Hermite’s equation in Section 4. Finally, a concise conclusion
is provided in Section 5.
2. A Basic Introduction to Time Scales
A time scale means an arbitrary nonempty closed subset of
the real numbers. The calculus of time scales was initiated by
Liu [3] in order to create a theory that can unify discrete and
continuous analysis.
Then, we introduce the delta derivative by starting to
define the forward and backward jump operators.
Definition 1. Let T be a time scale. For 𝑡 ∈ T, we define the
forward jump operator 𝜎 : T → T by
𝜎 (𝑡) := inf {𝑠 > 𝑡 | 𝑠 ∈ T} ,
(4)
while the backward jump operator 𝜌 : T → T by
𝑢ΔΔ (𝑡) + 𝑔 (𝑡) 𝑢Δ (𝑡) + 𝑓 (𝑡) 𝑢 (𝑡) = 0,
𝑡 ∈ 𝑞N ,
(2)
𝜌 (𝑡) := sup {𝑠 < 𝑡 | 𝑠 ∈ T} .
(5)
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Journal of Applied Mathematics
Definition 2. The graininess function 𝜇 : T → [0, ∞) is
defined by
𝜇 (𝑡) := 𝜎 (𝑡) − 𝑡.
∞
(6)
According to the basic definitions, we can give some
useful relationships concerning the delta derivative.
Let 𝑎 and 𝑞 be real numbers such that 0 < 𝑞 < 1. The
𝑞-shift factorial [4] is defined by
(𝑎; 𝑞)0 = 1,
at 𝑡0 that converges to 𝑓 near 𝑡0 ; that is, there exist coefficients
N
{𝑎𝑘 }∞
𝑘=0 and points 𝑐, 𝑑 ∈ 𝑞 such that 𝑐 < 𝑡0 < 𝑑 and
𝑛−1
(𝑎; 𝑞)𝑛 = ∏ (1 − 𝑎𝑞𝑘 ) ,
𝑛 = 1, 2, . . . , 𝑛.
𝑘=0
(7)
𝑓 (𝑡) = ∑ 𝑎𝑘 ℎ𝑘 (𝑡, 𝑡0 )
for all 𝑡 ∈ (𝑐, 𝑑) ∩ 𝑞N .
The production rule of two 𝑞-polynomials at 0 which will
be used to derive the series solution in following sections [3].
Theorem 6. Let ℎ𝑖 (𝑡, 0) and ℎ𝑗 (𝑡, 0) be two 𝑞-polynomials at
zero. One has
Assume 𝑓 : T → R is a function and 𝑡 ∈ T. The 𝑞derivative [5] at 𝑡 is defined by
𝑓 (𝑞𝑡) − 𝑓 (𝑡)
.
(𝑞 − 1) 𝑡
𝑓Δ (𝑡) =
(8)
(9)
𝑠
Hence, for each fixed 𝑠, the delta derivative of ℎ𝑘 with
respect to 𝑡 satisfies
ℎ𝑘Δ (𝑡, 𝑠) = ℎ𝑘−1 (𝑡, 𝑠) ,
𝑘 ≥ 1.
𝑖+𝑗−1
]
]=0 ∑𝜇=0
(11)
on 𝑞N [5].
Agarwal and Bohner [6] give a Taylor’s formula for
functions on a general time scale. On 𝑞N , the Taylor’s formula
can be rewritten as the following form.
Theorem 4. Let 𝑛 ∈ N. Suppose 𝑓 is 𝑛 times differentiable on
𝑞N . Let 𝛼, 𝑡 ∈ 𝑞N . One has
𝑛−1
𝑘
𝑓 (𝑡) = ∑ ℎ𝑘 𝑓Δ (𝛼) + ∫
𝑘=0
𝜌𝑛−1 (𝑡)
𝛼
𝑖−1
𝑖+𝑗−1
𝑡
= (∏
]
]=0 ∑𝜇=0
𝑞
)( ∏
𝜇
]=𝑖
Definition 5. A real-valued function 𝑓 : 𝑞N → R is said to be
𝑞-analytic at 𝑡0 if and only if there is a power series centered
,
(15)
𝑡
)
∑]𝜇=0 𝑞𝜇
𝑗−1
= ℎ𝑖 (𝑡, 0) (
∏]=0 ∑]𝜇=0 𝑞𝜇
𝑗−1
∏]=0
𝑗−1
= ℎ𝑖 (𝑡, 0) (∏
∑]𝜇=0
=
𝑞𝜇
𝑖+𝑗−1
) 𝑡𝑗 ( ∏
]=𝑖
𝑗−1 ]
𝑡
]
]=0 ∑𝜇=0
𝑞
1
)
∑]𝜇=0 𝑞𝜇
𝑖+𝑗−1
) (∏ ∑ 𝑞𝜇 ) ( ∏
𝜇
𝑞𝜇
𝑞𝜇
1
]
]=𝑖 ∑𝜇=0
]=0 𝜇=0
𝑗−1 ∑]
𝜇=0
ℎ𝑖 (𝑡, 0) ℎ𝑗 (𝑡, 0) (∏ ]+𝑖
]=0 ∑𝜇=0
𝑞𝜇
)
).
(16)
This implies that
𝑗−1 ∑]+𝑖
𝜇=0
]
∑
]=0 𝜇=0
ℎ𝑖 (𝑡, 0) ℎ𝑗 (𝑡, 0) = (∏
𝑗−1 (1
=∏
]=0
𝑛
Before developing the series solution method, we introduce the 𝑞-analytic function on 𝑞N .
𝑞𝜇
ℎ𝑖+𝑗 (𝑡, 0)
ℎ𝑛−1 (𝑡, 𝜎 (𝜏)) 𝑓Δ (𝜏) Δ𝜏.
(12)
(14)
we have
𝑘−1
𝑡 − 𝑠𝑞]
ℎ𝑘 (𝑡, 𝑠) = ∏ ] 𝑗
]=0 ∑𝑗=0 𝑞
𝑡
ℎ𝑖+𝑗 (𝑡, 0) = ∏
(10)
By computing the recurrence relation, the 𝑞-polynomials can
be represented as
ℎ𝑖+𝑗 (𝑡, 0) .
(𝑞; 𝑞)𝑗
Proof. Since
𝑡
ℎ𝑘+1 = ∫ ℎ𝑘 (𝜏, 𝑠) Δ𝜏.
ℎ0 (𝑡, 𝑠) = 1,
(𝑞𝑖+1 ; 𝑞)𝑗
ℎ𝑖 (𝑡, 0) ℎ𝑗 (𝑡, 0) =
A 𝑞-difference equation is an equation that contains 𝑞derivatives of a function defined on 𝑞N .
Definition 3. On the time scale T, the 𝑞-polynomials ℎ𝑘 (⋅, 𝑡0 ) :
T → R are defined recursively as follows:
(13)
𝑘=0
=
𝑞𝜇
𝑞𝜇
) ℎ𝑖+𝑗 (𝑡, 0)
− 𝑞𝜐+𝑖+1 )
(1 − 𝑞𝜐+1 )
(𝑞𝑖+1 ; 𝑞)𝑗
(𝑞; 𝑞)𝑗
ℎ𝑖+𝑗 (𝑡, 0)
(17)
ℎ𝑖+𝑗 (𝑡, 0) .
Proposition 7. Let ℎ𝑖 (𝑡) and ℎ𝑗 (𝑡) be any two 𝑞-polynomials.
One has
ℎ𝑖 (𝑡, 0) ℎ𝑗 (𝑡, 0) = ℎ𝑗 (𝑡, 0) ℎ𝑖 (𝑡, 0) .
(18)
Journal of Applied Mathematics
3
Proof. Without loss of generality, we suppose 𝑖 > 𝑗 and have
(𝑞𝑖+1 ; 𝑞)𝑗
(𝑞, 𝑞)𝑗
=
=
−
(𝑞𝑗+1 ; 𝑞)𝑖
𝑔 (𝑡) 𝑢Δ (𝑡)
(𝑞, 𝑞)𝑖
(1 − 𝑞𝑗+1 ) ⋅ ⋅ ⋅ (1 − 𝑞𝑖+𝑗 )
(1 − 𝑞) ⋅ ⋅ ⋅ (1 − 𝑞𝑖 )
−
(1 − 𝑞𝑖+1 ) ⋅ ⋅ ⋅ (1 − 𝑞𝑖+𝑗 )
(1 − 𝑞) ⋅ ⋅ ⋅ (1 −
𝑘=0
𝑙=0
∞ 𝑘
𝑘=0 𝑙=0
𝑘
(𝑞𝑙+1 , 𝑞)𝑘−𝑙
[
] ℎ𝑘 (𝑡, 0) ,
= ∑ ∑𝐺 (𝑙) 𝑈 (𝑘 + 1 − 𝑙)
(𝑞,
𝑞)
𝑘−𝑙
𝑘=0 𝑙=0
[
]
∞
(1 − 𝑞𝑖+1 ) ⋅ ⋅ ⋅ (1 − 𝑞𝑖+𝑗 ) (1 − 𝑞𝑗+1 ) ⋅ ⋅ ⋅ (1 − 𝑞𝑖 )
(1 − 𝑞) ⋅ ⋅ ⋅ (1 − 𝑞𝑗 ) (1 − 𝑞𝑗+1 ) ⋅ ⋅ ⋅ (1 − 𝑞𝑖 )
𝑓 (𝑡) 𝑢 (𝑡)
= 0.
(19)
This implies that
(𝑞, 𝑞)𝑗
∞
= ∑ ∑𝐺 (𝑙) 𝑈 (𝑘 + 1 − 𝑙) (ℎ𝑙 (𝑡, 0) ℎ𝑘−𝑙 (𝑡, 0))
𝑞𝑖 )
(𝑞𝑖+1 ; 𝑞)𝑗
∞
= [ ∑ 𝐺 (𝑘) ℎ𝑘 (𝑡, 0)] [∑𝑈 (𝑙 + 1) ℎ𝑙 (𝑡, 0)]
(1 − 𝑞) ⋅ ⋅ ⋅ (1 − 𝑞𝑗 )
(1 − 𝑞𝑗+1 ) ⋅ ⋅ ⋅ (1 − 𝑞𝑖+𝑗 )
−
we get
=
(𝑞𝑗+1 ; 𝑞)𝑖
(𝑞, 𝑞)𝑖
∞
∞
𝑘=0
𝑙=0
= [ ∑ 𝐹 (𝑘) ℎ𝑘 (𝑡, 0)] [∑𝑈 (𝑙) ℎ𝑙 (𝑡, 0)]
∞ 𝑘
.
(20)
= ∑ ∑𝐹 (𝑙) 𝑈 (𝑘 − 𝑙) (ℎ𝑙 (𝑡, 0) ℎ𝑘−𝑙 (𝑡, 0))
𝑘=0 𝑙=0
𝑘
(𝑞𝑙+1 , 𝑞)𝑘−𝑙
[
] ℎ𝑘 (𝑡, 0) .
= ∑ ∑𝐹 (𝑙) 𝑈 (𝑘 − 𝑙)
(𝑞, 𝑞)𝑘−𝑙
𝑘=0 𝑙=0
[
]
∞
Therefore, we have
ℎ𝑖 (𝑡, 0) ℎ𝑗 (𝑡, 0) = ℎ𝑗 (𝑡, 0) ℎ𝑖 (𝑡, 0)
(21)
by Theorem 6.
Substituting (24) and (26) into (22), we get the equation
3. Developing Series Solutions Method
Using the Taylor series on time scales, we develop a series
solution method for solving 𝑞-difference equations in this
section.
Consider a second-order 𝑞-difference equation
𝑢ΔΔ (𝑡) + 𝑔 (𝑡) 𝑢Δ (𝑡) + 𝑓 (𝑡) 𝑢 (𝑡) = 0,
𝑡 ∈ 𝑞N ,
(22)
where 𝑓 and 𝑔 are both 𝑞-analytic functions at 0 in the
interval (𝑐, 𝑑). Hence, there exist two sequences of coefficients
{𝐹(𝑘)} and {𝐺(𝑘)} such that
∞
𝑓 (𝑡) = ∑ 𝐹 (𝑘) ℎ𝑘 (𝑡, 0) ,
𝑘=0
∞
𝑔 (𝑡) = ∑ 𝐺 (𝑘) ℎ𝑘 (𝑡, 0) (23)
𝑈 (𝑘 + 2)
𝑘
[
(𝑞𝑙+1 , 𝑞)𝑘−𝑙
∞ [
[ +∑𝐺 (𝑙) 𝑈 (𝑘 + 1 − 𝑙)
(𝑞, 𝑞)𝑘−𝑙
∑[
𝑙=0
[
𝑙+1
[
𝑘
𝑘=0
(𝑞 , 𝑞)𝑘−𝑙
[
+∑𝐹 (𝑙) 𝑈 (𝑘 − 𝑙)
(𝑞, 𝑞)𝑘−𝑙
[
𝑙=0
𝑘
𝑈 (𝑘 + 2) + ∑𝐺 (𝑙) 𝑈 (𝑘 + 1 − 𝑙)
𝑘
+ ∑𝐹 (𝑙) 𝑈 (𝑘 − 𝑙)
(24)
𝑙=0
𝑘=0
Step 1. Since
∞
(𝑡) = ∑ 𝑈 (𝑘 + 2) ℎ𝑘 (𝑡, 0) ,
𝑘=0
(𝑞, 𝑞)𝑘−𝑙
(28)
= 0.
∞
𝑢Δ (𝑡) = ∑ 𝑈 (𝑘 + 1) ℎ𝑘 (𝑡, 0) ,
𝑢
(𝑞𝑙+1 , 𝑞)𝑘−𝑙
(𝑞, 𝑞)𝑘−𝑙
Step 3. Find all coefficients 𝑈(𝑘) in terms of the first two
coefficients 𝑈(0) and 𝑈(1), thus writing the 𝑞-series in the
form
by carrying out the following steps.
∞
(𝑞𝑙+1 , 𝑞)𝑘−𝑙
𝑙=0
∞
ΔΔ
]
Step 2. Set the coefficients of the power series equal to zero.
That gives a recurrence relation that relates later coefficients
in the power series (24) to the earlier ones. That is,
for all 𝑡 ∈ (𝑐, 𝑑) ∩ 𝑞N .
One can find a power series solution of the form
𝑘=0
]
]
]
] ℎ𝑘 (𝑡, 0) = 0.
]
]
]
(27)
𝑘=0
𝑢 (𝑡) = ∑ 𝑈 (𝑘) ℎ𝑘 (𝑡, 0) ,
(26)
∑ 𝑈 (𝑘) ℎ𝑘 (𝑡, 0) = 𝑈 (0) 𝑢1 (𝑡) + 𝑈 (1) 𝑢2 (𝑡) ,
(25)
(29)
𝑘=0
where 𝑢1 and 𝑢2 are two linearly independent 𝑞-series
solutions.
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Journal of Applied Mathematics
4. Applications
In this section, the series solution method is applied to
consider the 𝑞-Hermite’s equations with initial conditions.
Consider the 𝑞-Hermite’s equation of the form
𝑢
ΔΔ
Δ
(𝑡) − 𝑡𝑢 (𝑡) + 𝜆𝑢 (𝑡) = 0,
𝑡∈
𝑞N ,
(30)
Example 8. Consider the 𝑞-Hermite’s equation with 𝑞 = 1/2
of the form
𝑢ΔΔ − 𝑡𝑢Δ + 𝑢 = 0
with 𝑢(0) = 1 and 𝑢Δ = 0. Substituting (31) into (36) yields
𝑈 (2𝑙) = Π𝑙𝑖=1 [
Δ
with 𝑢(0) = 𝑎 and 𝑢 (0) = 𝑏.
Let
∞
𝑢 (𝑡) = ∑ 𝑈 (𝑘) ℎ𝑘 (𝑡, 0) ,
(31)
𝑘=0
then 𝑢(0) = 𝑎 = 𝑈(0) and 𝑢Δ (0) = 𝑏 = 𝑈(1). Applying (31)
into (30), we have
where 𝑘 = 1, 2, . . ..
This implies that
𝑈 (2𝑙) = [
(1 − 𝑞2(𝑙−1) )
(1 − 𝑞)
𝑙
= ∏[
(1 − 𝑞2(𝑖−1) )
(1 − 𝑞)
𝑖=1
𝑈 (2𝑙 + 1) = [
(1 − 𝑞2𝑙−1 )
𝑙
(1 − 𝑞)
= ∏[
− 𝜆] 𝑈 (2 (𝑙 − 1))
(33)
− 𝜆] 𝑈 (2𝑙 − 1)
(1 − 𝑞2𝑖−1 )
(1 − 𝑞)
𝑖=1
− 𝜆] 𝑎,
∞
1 2𝑖−3
𝑢 (𝑡) = 1 + ∑Π𝑙𝑖=1 1 − ( ) ℎ2𝑙 (𝑡)
2
𝑙=1
1
3
= 1 − ℎ2 (𝑡) − ℎ4 (𝑡) − ℎ6 (𝑡) − ⋅ ⋅ ⋅ .
2
8
− 𝜆] 𝑏.
𝑢 (𝑡) = 𝑎 (1 + ∑∏ [
(1 − 𝑞2(𝑖−1) )
𝑙=1 𝑖=1
∞
(1 − 𝑞)
𝑙
+ 𝑏 (ℎ1 (𝑡) + ∑ ∏ [
𝑙=1 𝑖=1
− 𝜆] ℎ2𝑙 (𝑡))
(1 − 𝑞2𝑖−1 )
(1 − 𝑞)
(38)
5. Conclusion
One area which is the lack of development is the theory of
series solutions on 𝑞-difference equations. In this paper, we
present the formula for the multiplicity of two 𝑞-polynomials
at 0. The purpose is to provide the basic mechanics for finding the series solutions of linear homogeneous 𝑞-difference
equation. As an example we consider series solution of the 𝑞Hermite’s equation of the form 𝑢ΔΔ − 𝑡𝑢Δ + 𝜆𝑢 = 0, with the
initial conditions 𝑢(0) = 𝑎 and 𝑢Δ (0) = 𝑏. Using the presented
method, the series solution of the Hermite’s equation can
be obtained iteratively. In future studies, we would apply
the presented method to find the series solution of other 𝑞difference equations on 𝑞N .
References
Hence, we get
∞ 𝑙
1 − (1/2)2(𝑖−1)
1 2𝑖−3
− 1] = Π𝑙𝑖=1 [1 − ( ) ]
2
(1/2)
(37)
and 𝑈(2𝑙 + 1) = 0 which implies that
𝑈 (2) = −𝜆𝑈 (0) ,
(𝑞𝑘 ; 𝑞)1
(1 − 𝑞𝑘 )
𝑈 (𝑘 + 2) = [
− 𝜆] 𝑈 (𝑘) = [
− 𝜆] 𝑈 (𝑘) ,
(𝑞; 𝑞)1
(1 − 𝑞)
[
]
(32)
(36)
− 𝜆] ℎ2𝑙+1 (𝑡))
= 𝑎𝑢1 (𝑡) + 𝑏𝑢2 (𝑡) .
(34)
By computing the Wronskian of 𝑢1 and 𝑢2 at 0, we get
𝑊 [𝑢1 , 𝑢2 ] (0) = 𝑢1 (0) 𝑢2Δ (0) − 𝑢2 (0) 𝑢1Δ (0) = 1 ≠ 0. (35)
This implies that 𝑢1 and 𝑢2 are two linearly independent
solutions.
[1] B. D. Haile and L. M. Hall, “Polynomial and series solutions
of dynamic equations on time scales,” Dynamic Systems and
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[2] D. Mozyrska and E. Pawłuszewicz, “Hermite’s equations on time
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