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
(1)qℕ={qn∣n∈ℕ}, qℕ¯=qℕ∪{0},
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
(2)uΔΔ(t)+g(t)uΔ(t)+f(t)u(t)=0, t∈qℕ¯,
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
(3)uΔΔ(t)-tuΔ(t)+λu(t)=0, t∈qℕ¯
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 q-Hermite'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
(4)σ(t):=inf{s>t∣s∈𝕋},
while the backward jump operator ρ:𝕋→𝕋 by
(5)ρ(t):=sup{s<t∣s∈𝕋}.
Definition 2.
The graininess function μ:𝕋→[0,∞) is defined by
(6)μ(t):=σ(t)-t.
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
(7)(a;q)0=1, (a;q)n=∏k=0n-1(1-aqk), n=1,2,…,n.
Assume f:𝕋→ℝ is a function and t∈𝕋. The q-derivative [5] at t is defined by
(8)fΔ(t)=f(qt)-f(t)(q-1)t.
A q-difference equation is an equation that contains q-derivatives 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:
(9)h0(t,s)=1, hk+1=∫sthk(τ,s)Δτ.
Hence, for each fixed s, the delta derivative of hk with respect to t satisfies
(10)hkΔ(t,s)=hk-1(t,s), k≥1.
By computing the recurrence relation, the q-polynomials can be represented as
(11)hk(t,s)=∏ν=0k-1t-sqν∑j=0νqj
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
(12)f(t)=∑k=0n-1hkfΔk(α)+∫αρn-1(t)hn-1(t,σ(τ))fΔn(τ)Δτ.
Before developing the series solution method, we introduce the q-analytic function on qℕ¯.
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 at t0 that converges to f near t0; that is, there exist coefficients {ak}k=0∞ and points c,d∈qℕ¯ such that c<t0<d and
(13)f(t)=∑k=0∞akhk(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
(14)hi(t,0)hj(t,0)=(qi+1;q)j(q;q)jhi+j(t,0).
Proof.
Since
(15)hi+j(t,0)=∏ν=0i+j-1t∑μ=0νqμ,
we have
(16)hi+j(t,0) =(∏ν=0i-1t∑μ=0νqμ)(∏ν=ii+j-1t∑μ=0νqμ) =hi(t,0)(∏ν=0j-1∑μ=0νqμ∏ν=0j-1∑μ=0νqμ)tj(∏ν=ii+j-11∑μ=0νqμ) =hi(t,0)(∏ν=0j-1t∑μ=0νqμ)(∏ν=0j-1∑μ=0νqμ)(∏ν=ii+j-11∑μ=0νqμ) =hi(t,0)hj(t,0)(∏ν=0j-1∑μ=0νqμ∑μ=0ν+iqμ).
This implies that
(17)hi(t,0)hj(t,0)=(∏ν=0j-1∑μ=0ν+iqμ∑μ=0νqμ)hi+j(t,0)=∏ν=0j-1(1-qυ+i+1)(1-qυ+1)hi+j(t,0)=(qi+1;q)j(q;q)jhi+j(t,0).
Proposition 7.
Let hi(t) and hj(t) be any two q-polynomials. One has
(18)hi(t,0)hj(t,0)=hj(t,0)hi(t,0).
Proof.
Without loss of generality, we suppose i>j and have(19)(qi+1;q)j(q,q)j-(qj+1;q)i(q,q)i =(1-qj+1)⋯(1-qi+j)(1-q)⋯(1-qi)-(1-qi+1)⋯(1-qi+j)(1-q)⋯(1-qj) =(1-qj+1)⋯(1-qi+j)(1-q)⋯(1-qi) -(1-qi+1)⋯(1-qi+j)(1-qj+1)⋯(1-qi)(1-q)⋯(1-qj)(1-qj+1)⋯(1-qi) =0.
This implies that
(20)(qi+1;q)j(q,q)j=(qj+1;q)i(q,q)i.
Therefore, we have
(21)hi(t,0)hj(t,0)=hj(t,0)hi(t,0)
by Theorem 6.
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
(22)uΔΔ(t)+g(t)uΔ(t)+f(t)u(t)=0, t∈qℕ¯,
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
(23)f(t)=∑k=0∞F(k)hk(t,0), g(t)=∑k=0∞G(k)hk(t,0)
for all t∈(c,d)∩qℕ¯.
One can find a power series solution of the form
(24)u(t)=∑k=0∞U(k)hk(t,0),
by carrying out the following steps.
Step 1.
Since
(25)uΔ(t)=∑k=0∞U(k+1)hk(t,0),uΔΔ(t)=∑k=0∞U(k+2)hk(t,0),
we get
(26)g(t)uΔ(t) =[∑k=0∞G(k)hk(t,0)][∑l=0∞U(l+1)hl(t,0)] =∑k=0∞ ∑l=0kG(l)U(k+1-l)(hl(t,0)hk-l(t,0)) =∑k=0∞[∑l=0kG(l)U(k+1-l)(ql+1,q)k-l(q,q)k-l]hk(t,0),f(t)u(t) =[∑k=0∞F(k)hk(t,0)][∑l=0∞U(l)hl(t,0)] =∑k=0∞ ∑l=0kF(l)U(k-l)(hl(t,0)hk-l(t,0)) =∑k=0∞[∑l=0kF(l)U(k-l)(ql+1,q)k-l(q,q)k-l]hk(t,0).
Substituting (24) and (26) into (22), we get the equation
(27)∑k=0∞[U(k+2) +∑l=0kG(l)U(k+1-l)(ql+1,q)k-l(q,q)k-l +∑l=0kF(l)U(k-l)(ql+1,q)k-l(q,q)k-l]hk(t,0)=0.
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,
(28)U(k+2)+∑l=0kG(l)U(k+1-l)(ql+1,q)k-l(q,q)k-l +∑l=0kF(l)U(k-l)(ql+1,q)k-l(q,q)k-l=0.
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
(29)∑k=0∞U(k)hk(t,0)=U(0)u1(t)+U(1)u2(t),
where u1 and u2 are two linearly independent q-series solutions.
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
(30)uΔΔ(t)-tuΔ(t)+λu(t)=0, t∈qℕ¯,
with u(0)=a and uΔ(0)=b.
Let
(31)u(t)=∑k=0∞U(k)hk(t,0),
then u(0)=a=U(0) and uΔ(0)=b=U(1). Applying (31) into (30), we have
(32)U(2)=-λU(0),U(k+2)=[(qk;q)1(q;q)1-λ]U(k)=[(1-qk)(1-q)-λ]U(k),
where k=1,2,….
This implies that
(33)U(2l)=[(1-q2(l-1))(1-q)-λ]U(2(l-1))=∏i=1l[(1-q2(i-1))(1-q)-λ]a,U(2l+1)=[(1-q2l-1)(1-q)-λ]U(2l-1)=∏i=1l[(1-q2i-1)(1-q)-λ]b.
Hence, we get
(34)u(t)=a(1+∑l=1∞∏i=1l[(1-q2(i-1))(1-q)-λ]h2l(t))+b(h1(t)+∑l=1∞ ∏i=1l[(1-q2i-1)(1-q)-λ]h2l+1(t))=au1(t)+bu2(t).
By computing the Wronskian of u1 and u2 at 0, we get
(35)W[u1,u2](0)=u1(0)u2Δ(0)-u2(0)u1Δ(0)=1≠0.
This implies that u1 and u2 are two linearly independent solutions.
Example 8.
Consider the q-Hermite's equation with q=1/2 of the form
(36)uΔΔ-tuΔ+u=0
with u(0)=1 and uΔ=0. Substituting (31) into (36) yields
(37)U(2l)=Πi=1l[1-(1/2)2(i-1)(1/2)-1]=Πi=1l[1-(12)2i-3]
and U(2l+1)=0 which implies that
(38)u(t)=1+∑l=1∞Πi=1l1-(12)2i-3h2l(t)=1-h2(t)-12h4(t)-38h6(t)-⋯.