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Fractional q-symmetric calculus on a time scale
Advances in Difference Equations volume 2017, Article number: 166 (2017)
Abstract
In this paper, the definitions of q-symmetric exponential function and q-symmetric gamma function are presented. By a q-symmetric exponential function, we shall illustrate the Laplace transform method and define and solve several families of linear fractional q-symmetric difference equations with constant coefficients. We also introduce a q-symmetric analogue Mittag-Leffler function and study q-symmetric Caputo fractional initial value problems. It is hoped that our work will provide foundation and motivation for further studying of fractional q-symmetric difference systems.
1 Introduction
The q-calculus is not of recent appearance. It was initiated in the twenties of the last century. However, it has gained considerable popularity and importance during the last three decades or so. This is due to its distinguished applications in numerous diverse fields of physics such as cosmic strings and black holes [1], conformal quantum mechanics [2], nuclear and high energy physics [3], just to name a few. Early developments for q-fractional calculus can be found in the work of Al-Salam and co-authors [4, 5] or Agarwal [6]. A q-Laplace transform method has been developed by Abdi [7] and applied to q-difference equations [8, 9]. Moreover, there is currently much activity to reexamine and further develop the q-special functions. Notable early work includes the work of Jackson [10–14], Hahn [15, 16] and Agarwal [17]. We also refer the reader to more recent articles [18–26] and [27, 28].
The q-symmetric quantum calculus has proven to be useful in several fields, in particular in quantum mechanics [29]. As noticed in [30], consistently with the q-deformed theory, the standard q-symmetric integral must be generalized to the basic integral defined. Recently, we introduced basic concepts of fractional q-symmetric integral and derivative operators in [31]. The basic theory of q-symmetric quantum calculus operators need be explored. The object of this paper is to further develop the theory of fractional q-symmetric calculus. First, the definitions of q-symmetric exponential function and q-symmetric gamma function are presented. Second, we give a fractional q-symmetric transform method. We then define some q-symmetric difference equations and apply the q-symmetric transform method to obtain solutions. Finally, we study a Caputo q-fractional initial value problem and give a q-analogue Mittag-Leffler function.
2 The q-symmetric gamma and the q-symmetric exponential functions
Let \(t_{0}\in \mathbb{R} \) and define
If there is no confusion concerning \(t_{0}\), we shall denote \(\mathbb{T}_{t_{0}}\) by \(\mathbb{T}\).
The basic q-symmetric integrals of \(f: \mathbb{T}\longrightarrow \mathbb{R}\) are defined through the relations
For \(a\in \mathbb{T}\),
For a function \(f:\mathbb{T}\longrightarrow \mathbb{R}\), the q-symmetric derivative is defined by
and the q-symmetric derivatives of higher order are defined by
As for q-symmetric derivatives, we can define an operator \(\widetilde{I}^{n}_{q}\) by
For operators defined in this manner, the following is valid:
The formula for q-symmetric integration by parts is
Definition 2.1
The q-symmetric factorial is defined in the following way. If n is a nonnegative integer, then
Their natural expansions to reals are
Next, for a real parameter \(q\in \mathbb{R}^{+}\setminus \{1\}\), we introduce a q-real number \(\overline{[a]}_{q}\) by
We shall state several properties of the q-symmetric factorial function, each property is verified using the definition and a straightforward calculation.
Theorem 2.1
For a q-symmetric factorial function, we have
-
(i)
\(\overline{(t-s)}^{(\beta +\nu)}=\overline{(t-s)}^{(\beta)}\overline{ \bigl(t-q ^{2\beta }s\bigr)}^{(\nu)}\),
-
(ii)
\(\overline{(at-as)}^{(\beta)}=a^{\beta }\overline{(t-s)}^{( \beta)}\),
-
(iii)
\({} _{t}\widetilde{D}_{q}\overline{(t-s)}^{(\beta)}= \overline{[ \beta ]}_{q}\overline{\bigl(q^{-1}t-s \bigr)}^{(\beta -1)}\),
-
(iv)
\({} _{s}\widetilde{D}_{q}\overline{ \bigl(t-q^{-1}s\bigr)}^{(\beta)}=-\overline{[ \beta ]}_{q}\overline{(t-s)}^{(\beta -1)}\).
Definition 2.2
The q-symmetric exponential function is defined as
Note that \(\overline{e}_{q}(q^{-1})=0\) and
We are now in a position to give the integral representation of the q-symmetric gamma function. Let \(\alpha \in \mathbb{R}\setminus \{\ldots,-2,-1,0\}\). Define the q-symmetric gamma function by
For an integer n, we denote
Lemma 2.1
For \(\alpha \in \mathbb{R}\),
For any positive integer k,
Proof
By (2.5), (2.7), (2.8), we can get
 □
Remark 2.1
In [31], authors introduced the q-symmetric gamma function as
To see that \(\widetilde{\Gamma }_{q}(\alpha)=\widetilde{\Gamma ^{*}} _{q}(\alpha)\), we use the following formula given by Atici and Eloe in [9]:
where \(e_{q}(t)=\prod_{n=0}^{\infty }(1-q^{n}t)\).
In fact
hence
3 Fractional q-symmetric integral and derivative
We now introduce the fractional q-symmetric integral operator (see [31])
where
Lemma 3.1
[31]
Let \(\alpha,\beta \in \mathbb{R}^{+}\). The fractional q-symmetric integration has the following semigroup property:
Lemma 3.2
[31]
For \(\alpha \in \mathbb{R}^{+}\), the following identity is valid:
We define the fractional q-symmetric derivative of Riemann-Liouville type of a function \(f(x)\) by
where \([\alpha ]\) denotes the smallest integer greater or equal to α.
Lemma 3.3
[31]
For \(\alpha \in \mathbb{R}\setminus \mathbb{N}_{0}\), the following is valid:
Lemma 3.4
[31]
For \(\alpha \in \mathbb{R}\setminus \mathbb{N}_{0}\), the following is valid:
Lemma 3.5
[31]
For \(\alpha \in \mathbb{R}\setminus \mathbb{N}_{0}\), the following is valid:
Lemma 3.6
[31]
Let \(\alpha \in (N-1,N]\). Then, for some constants \(c_{i}\in \mathbb{R}\), \(i=1,2,\ldots,N\), the following equality holds:
If we change the order of operators, we can introduce another type of fractional q-derivative.
The fractional q-symmetric derivative of Caputo type is
where \([\alpha ]\) denotes the smallest integer greater or equal to α.
Lemma 3.7
[31]
For \(\alpha \in \mathbb{R}\setminus \mathbb{N}_{0}\), and \(x>0\), the following is valid:
Lemma 3.8
[31]
For \(\alpha \in \mathbb{R}\setminus \mathbb{N}_{0}\) and \(x>0\), the following is valid:
Lemma 3.9
[31]
Let \(\alpha \in (N-1,N]\). Then, for some constants \(c_{i}\in \mathbb{R}\), \(i=0,1,\ldots,N-1\), the following equality holds:
Lemma 3.10
Let \(f:\mathbb{T}\times \mathbb{T}\longrightarrow \mathbb{R}\), then the following identity holds:
Proof
 □
Theorem 3.1
If \(f(t)\) is defined and finite, then for \(\nu >0\) with \(N-1<\nu \leq N\),
Proof
Using Lemma 3.10 and Theorem 2.1, we have
 □
4 The q-symmetric Laplace transform
For convenience, we need some preliminaries.
Let \(q^{2}=\overline{q}\), then we have
The basic q-integrals are defined by
Definition 4.1
[9] q-beta function
For any \(x,y>0\), \(B_{q}(x,y)=\int_{0}^{1}t^{x-1}(1-qt)^{(y-1)}\,d_{q}t\).
Recall that
Therefore,
Similar to q-beta function, we shall define the q-symmetric beta function.
Definition 4.2
q-symmetric beta function
For any \(x,y>0\), \(\widetilde{B}_{q}(x,y)=\int_{0}^{1}(q^{-1}t)^{x-1} \overline{(1-t)}^{(y-1)}\,\widetilde{d}_{q}t\).
Lemma 4.1
For any \(x,y>0\), the following equality is valid:
Proof
By (2.1), (4.2), (4.3) and Definition 4.2, we have
 □
Lemma 4.2
[31]
For \(\alpha \in \mathbb{R}^{+}\setminus \mathbb{N}_{0}\), \(\lambda \in (-1,\infty)\), the following is valid:
-
(i)
\(\widetilde{I}^{\alpha }_{q,0}x^{\lambda }= \frac{ \widetilde{\Gamma }_{q}(\lambda +1)}{\widetilde{\Gamma }_{q}(\lambda +\alpha +1)}q^{{\bigl({\scriptsize\begin{matrix}{} \alpha \cr 2 \end{matrix}}\bigr) }+\lambda \alpha }x^{\lambda +\alpha }\),
-
(ii)
\(\widetilde{D}^{\alpha }_{q,0}x^{\lambda }= \frac{ \widetilde{\Gamma }_{q}(\lambda +1)}{\widetilde{\Gamma }_{q}(\lambda -\alpha +1)}q^{{\bigl({\scriptsize\begin{matrix}{} \alpha \cr 2 \end{matrix}}\bigr) }-\lambda \alpha }x^{\lambda -\alpha }\).
We shall define a q-symmetric Laplace transform as follows:
Lemma 4.3
For any \(\alpha \in \mathbb{R}\setminus \{\ldots,-2,-1,0\}\),
Proof
 □
We now turn our attention to a shift theorem for the q-symmetric Laplace transform. First note the following identity.
Lemma 4.4
Let a be any real number. Then
Let n denote a positive integer. Then
Proof
Note that
and
so for \(n=0\),
The proof proceeds by induction. Let \(n\geq 1\) be an integer. Note that
we have
i.e.,
so
 □
Next, let \(F_{1}(t)=t^{\mu }\), \(F_{2}(t)=t^{\nu -1}\). Define \(F_{2}[t]= \overline{(t-rt)}^{(\nu -1)}\) and the convolution
where \(\nu \in \mathbb{R}\setminus \{\ldots,-2,-1,0\}\), \(\mu \in (-1,+ \infty)\).
By the power rule, we have
In fact, by (3.1), Lemma 4.2, we have
Now we simply apply Lemma 4.3 to each of \(F_{1}\), \(F_{2}\), \(F_{1}\ast F _{2}\) and obtain a convolution theorem.
Theorem 4.1
Proof
Similarly,
Thus (4.4) holds. □
The convolution theorem will be valid for functions \(F_{1}\) representing linear sums of functions of the form \(t^{\mu }\). Clearly, μ is not necessary an integer.
Corollary 4.1
Let \(F_{1}\) be an analytic function, and let \(F_{2}=t^{\nu -1}\) on \(\mathbb{T}\setminus \{0\}\). Then Theorem 4.1 holds.
We now obtain some of the standard properties for the \(\widetilde{L} _{q}\)-transform.
Lemma 4.5
Assume \(f=F_{1}\) is of the type such that (4.4) is valid. Then
-
(i)
\(\widetilde{L}_{q}\bigl\{\widetilde{I}_{q}^{\nu }f \bigl(q^{-\nu }t\bigr)\bigr\}(s)=\frac{q ^{\bigl({\scriptsize\begin{matrix}{} -\nu \cr 2 \end{matrix}}\bigr) } (1-q^{2})^{\nu }}{s^{\nu }}\widetilde{L}_{q} \bigl\{f(t)\bigr\}(s)\).
-
(ii)
\(\widetilde{L}_{q}\{\widetilde{D}_{q}^{\nu }f(q^{\nu }t)\}(s)=\frac{q ^{\bigl({\scriptsize\begin{matrix}{} \nu \cr 2 \end{matrix}}\bigr) } (1-q^{2})^{\nu }}{s^{\nu }}\widetilde{L}_{q}\{f(t)\}(s)\).
Proof
(i) Note that
we have
Similarly, by Theorem 3.1, we may easily see that (ii) holds. □
For the next set of properties, first note that
and
Thus
It follows by induction that if m denotes a positive integer, then
By Lemma 4.5 and (4.5), we may easily obtain the following Theorem 4.2.
Theorem 4.2
If f is an analytic function on \(\mathbb{T} \setminus \{0\}\), \(\nu \in (N-1,N]\), then we have
Theorem 4.3
If \(\widetilde{L}_{q}\{f(t)\}(s)=F(s)\), then \(\widetilde{L}_{q}\{f(at)\}(s)=\frac{1}{a}F(\frac{s}{a})\).
Proof
 □
Example 4.1
Consider the following fractional q-symmetric difference equations:
-
(a)
\(\widetilde{D}^{3/2}_{q}y(q^{3/2}t)=0\) for \(\mathbb{T} \setminus \{0\}\).
-
(b)
\(\widetilde{D}_{q}\widetilde{D}^{1/2}_{q}y(q^{3/2}t)=0\) for \(\mathbb{T}\setminus \{0\}\). Assume that \(\widetilde{D} ^{1/2}_{q}y(0)\) is defined and finite.
-
(c)
\(\widetilde{D}^{2}_{q}\widetilde{I}^{1/2}_{q}y(q^{3/2}t)=0\) for \(\mathbb{T}\setminus \{0\}\). Assume that \(\widetilde{I} ^{1/2}_{q,0}y(0)\), \(\widetilde{D}_{q}\widetilde{I}^{1/2}_{q,0}y(0)\) is defined and finite.
Note that each equation given in each of (a), (b) and (c) is not equivalent since our method requires knowledge of the fractional derivatives or integrals of the solution defined at zero as well.
We search for analytic solutions on \(\mathbb{T}\setminus \{0\}\) for each equation by using a q-symmetric Laplace transform.
For part (a), if we take the Laplace transform of each side of the equation, then by Theorem 4.2 we have \(y(t)=0\).
For part (b), if we take the Laplace transform of each side of the equation and use the properties of the q-symmetric Laplace transform, we obtain
By using Theorem 4.2, we have
For part (c), if we take the Laplace transform of each side of the equation and use the properties of the q-symmetric Laplace transform, we obtain
By using Lemma 4.3, we can get
Example 4.2
Consider the following fractional q-symmetric difference equation:
Applying the Laplace transform to each side of the equation, we can get
Example 4.3
Consider the problem
Applying the Laplace transform to each side of the equation, we can get
5 A q-symmetric fractional initial problem and q-symmetric Mittag-Leffler function
The following identity, which is useful for transforming q-symmetric fractional difference equations into q-symmetric fractional integrals, will be our key in this section.
Example 5.1
Let \(0<\alpha \leq 1\) and consider the q-symmetric fractional difference equation
If we apply \(\widetilde{I}_{q,0}^{\alpha }\) to the equation, then by Lemma 3.9 we see that
where \(g(t)=f(q^{ -\alpha }t)\).
To obtain an explicit clear solution, we apply the method of successive approximation. Set \(y_{0}(t)=a_{0}\) and
For \(m=1\), we have by the power formula Theorem 2.1
Since
we have
If we proceed inductively and let \(m\longrightarrow \infty \), we obtain the solution.
If we set \(\alpha =1\), \(\lambda =1\), \(a_{0}=0\), \(f(t)=0\) and note Remark 2.1, we come to the q-symmetric exponential formula
where \(\overline{(q)}_{n}=(1-q^{2})(1-q^{4})\cdots (1-q^{2k})\) is the q-symmetric Pochhammer symbol.
If compared with the classical case, the above example suggests the following q-symmetric analogue of the Mittag-Leffler function.
Definition 5.1
For \(z,z_{0}\in \mathbb{C}\) and \(\mathfrak{R}( \alpha)>0\), the q-symmetric Mittag-Leffler function is defined by
When \(\beta =1\), we simply use \(_{q}E_{\alpha }(\lambda,z-z_{0}):=_{q}E _{\alpha,1}(\lambda,z-z_{0})\).
According to Definition 5.1 above, the solution of the q-symmetric difference equation in Example 5.1 is expressed by
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Acknowledgements
The authors would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC project (No. 11161049).
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CH and MS worked together in the derivation of the mathematical results. All authors read and approved the final manuscript.
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Sun, M., Hou, C. Fractional q-symmetric calculus on a time scale. Adv Differ Equ 2017, 166 (2017). https://doi.org/10.1186/s13662-017-1219-x
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DOI: https://doi.org/10.1186/s13662-017-1219-x
MSC
- 92B20
- 68T05
- 39A11
- 34K13
Keywords
- fractional q-symmetric exponential function
- fractional q-symmetric gamma function
- Laplace transform
- initial value problem