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On homogeneous second order linear general quantum difference equations
Journal of Inequalities and Applications volume 2017, Article number: 198 (2017)
Abstract
In this paper, we prove the existence and uniqueness of solutions of the βCauchy problem of second order βdifference equations
\(a_{0}(t)\neq0\), in a neighborhood of the unique fixed point \(s_{0}\) of the strictly increasing continuous function β, defined on an interval \(I\subseteq{\mathbb{R}}\). These equations are based on the general quantum difference operator \(D_{\beta}\), which is defined by \(D_{\beta}{f(t)}= (f(\beta(t))f(t) )/ (\beta(t)t )\), \(\beta(t)\neq t\). We also construct a fundamental set of solutions for the second order linear homogeneous βdifference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the EulerCauchy βdifference equation.
1 Introduction
Quantum calculus allows us to deal with sets of nondifferentiable functions by substituting the classical derivative by a difference operator. Nondifferentiable functions are used to describe many important physical phenomena. Quantum calculus has a lot of applications in different mathematical areas such as the calculus of variations, orthogonal polynomials, basic hypergeometric functions, economical problems with a dynamic nature, quantum mechanics and the theory of scale relativity; see, e.g., [1–9]. The general quantum difference operator \(D_{\beta}\) is defined, in [10, p.6], by
where \(f:I\rightarrow\mathbb{X}\) is a function defined on an interval \(I\subseteq{\mathbb{R}}\), \(\mathbb{X}\) is a Banach space and \(\beta :I\rightarrow I\) is a strictly increasing continuous function defined on I, which has only one fixed point \(s_{0}\in{I}\) and satisfies the inequality: \((ts_{0})(\beta(t)t)\leq0\) for all \(t\in I\). The function f is said to be βdifferentiable on I, if the ordinary derivative \({f'}\) exists at \(s_{0}\). The βdifference operator yields the Hahn difference operator when \(\beta(t)=qt+\omega\), \(\omega>0\), \(q \in(0,1)\), and the Jackson qdifference operator when \(\beta(t)=qt\), \(q \in(0,1)\); see [11–16]. In [10], [17, Chapter 2], the definition of the βderivative, the βintegral, the fundamental theorem of βcalculus, the chain rule, Leibniz’s formula and the mean value theorem were introduced. In [18], the βexponential, βtrigonometric and βhyperbolic functions were presented. In [19], the existence and uniqueness of solutions of the βinitial value problem of the first order were established. In addition, an expansion form of the βexponential function was deduced.
This paper is devoted for deducing some results of the solutions of the homogeneous second order linear βdifference equations which are based on \(D_{\beta}\). In Section 2, we introduce the needed preliminaries of the βcalculus from [10, 17–19]. In Section 3, we prove the existence and uniqueness of solutions of the βCauchy problem of second order βdifference equations in a neighborhood of \(s_{0}\). We also construct a fundamental set of solutions for the second order linear homogeneous βdifference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the EulerCauchy βdifference equation. Throughout this paper, J is a neighborhood of the unique fixed point \(s_{0}\) of β and \(\mathbb{X}\) is a Banach space. If f is βdifferentiable two times over I, then the second order derivative of f is denoted by \(D_{\beta}^{2}f=D_{\beta}(D_{\beta}f)\). Furthermore, \(S(y_{0}, b)=\{y\in \mathbb{X}:\yy_{0}\\leq b\}\) and the rectangle \(R=\{(t,y)\in{{I}\times \mathbb{X}}:ts_{0}\leq{a},\yy_{0}\\leq{b}\}\), where a, b are fixed positive real numbers.
2 Preliminaries
In this section, we present some needed results associated with the βcalculus from [10, 17–19].
Lemma 2.1
The following statements are true:

(i)
The sequence of functions \(\{\beta^{k}(t)\}_{k=0}^{\infty}\) converges uniformly to the constant function \(\hat{\beta}(t):=s_{0}\) on every compact interval \(V \subseteq I\) containing \(s_{0}\).

(ii)
The series \(\sum_{k=0}^{\infty}\beta^{k}(t)\beta^{k+1}(t)\) is uniformly convergent to \(ts_{0} \) on every compact interval \(V \subseteq I\) containing \(s_{0}\).
Lemma 2.2
If \(f:I\rightarrow\mathbb{X}\) is a continuous function at \(s_{0}\), then the sequence \(\{f(\beta^{k}(t))\}_{k=0}^{\infty}\) converges uniformly to \(f(s_{0})\) on every compact interval \(V\subseteq I\) containing \(s_{0}\).
Theorem 2.3
If \(f:I\rightarrow\mathbb{X}\) is continuous at \(s_{0}\), then the series \(\sum_{k=0}^{\infty}\ (\beta^{k}(t)\beta^{k+1}(t) ) f(\beta ^{k}(t))\\) is uniformly convergent on every compact interval \(V \subseteq I\) containing \(s_{0}\).
Lemma 2.4
Let \(f:{I}\rightarrow\mathbb{X}\) be βdifferentiable and \({D}_{\beta}f(t)=0\) for all \(t\in{I}\). Then \(f(t)=f(s_{0})\) for all \(t\in{I}\).
Theorem 2.5
Assume that \(f:{I}\rightarrow\mathbb {X}\) and \(g:{I}\rightarrow\mathbb{R}\) are βdifferentiable functions on I. Then:

(i)
the product \(fg:I\rightarrow\mathbb{X}\) is βdifferentiable on I and
$$\begin{aligned} {D}_{\beta}(fg) (t) &=\bigl({D}_{\beta}f(t) \bigr)g(t)+f\bigl(\beta(t)\bigr){D}_{\beta}g(t) \\ & =\bigl({D}_{\beta}f(t)\bigr)g\bigl(\beta(t)\bigr)+f(t){D}_{\beta}g(t), \end{aligned} $$ 
(ii)
\(f/g\) is βdifferentiable at t and
$${D}_{\beta} ({f}/{g} ) (t)=\frac{({D}_{\beta }f(t))g(t)f(t){D}_{\beta}g(t)}{g(t)g(\beta(t))}, $$provided that \(g(t)g(\beta(t))\neq{0}\).
Theorem 2.6
Assume \(f:{I}\to\mathbb{X}\) is continuous at \(s_{0}\). The function F defined by
is a βantiderivative of f with \(F(s_{0})=0\). Conversely, a βantiderivative F of f vanishing at \(s_{0}\) is given by (2.1).
Definition 2.7
Let \(f:{I}\rightarrow{\mathbb{X}}\) and \(a,b\in{I}\). The βintegral of f from a to b is
where
provided that the series converges at \(x=a\) and \(x=b\). f is called βintegrable on I if the series converges at a and b for all \(a,b\in{I}\). Clearly, if f is continuous at \(s_{0}\in{I}\), then f is βintegrable on I.
Definition 2.8
The βexponential functions \(e_{p,\beta}(t)\) and \(E_{p,\beta}(t)\) are defined by
and
where \(p:I \rightarrow\mathbb{C}\) is a continuous function at \(s_{0}\) and both infinite products are convergent to a nonzero number for every \(t\in I\) and \(e_{p,\beta}(t)=\frac {1}{E_{p,\beta}(t)}\).
It is worth mentioning that both products in (2.2) and (2.3) are convergent since \(\sum_{k=0}^{\infty}  p(\beta^{k}(t)) (\beta^{k}(t)\beta ^{k+1}(t) ) \) is uniformly convergent. See [18, Definition 2.1].
Theorem 2.9
The βexponential functions \(e_{p,\beta}(t)\) and \(E_{p,\beta }(t)\) are, respectively, the unique solutions of the βinitial value problems:
Theorem 2.10
Assume that \(p,q:I\rightarrow\mathbb {C}\) are continuous functions at \(s_{0}\in I\). The following properties are true:

(i)
\(\frac{1}{e_{p,\beta}(t)}=e_{p/[1+(\beta(t)t)p]}(t)\),

(ii)
\(e_{p,\beta}(t)e_{q,\beta}(t)=e_{p+q+(\beta(t)t)pq}(t)\),

(iii)
\(e_{p,\beta}(t)/e_{q,\beta}(t)=e_{(pq)/[1+(\beta(t)t)q]}(t)\).
Definition 2.11
The βtrigonometric functions are defined by
Theorem 2.12
For all \(t\in I\). The following relation holds true:
Theorem 2.13
Assume that the function \(f:R\rightarrow {\mathbb{X}}\) is continuous at \((s_{0},y_{0})\in{R}\) and satisfies the Lipschtiz condition (with respect to y)
Then the βinitial value problem \(D_{\beta}{y(t)}=f(t,y)\), \(y(s_{0})=y_{0}\), \(t\in{I}\) has a unique solution on \([s_{0}\delta,s_{0}+\delta]\), where L is a positive constant and \(\delta=\min\{a,\frac{b}{Lb+M},\frac{\rho}{L}\}\) with \(M=\sup_{(t,y)\in{R}}\f(t,y)\<\infty\), \(\rho\in(0,1)\).
3 Main results
In this section, we prove the existence and uniqueness of solutions of the βCauchy problem of second order βdifference equations in a neighborhood of \(s_{0}\). Furthermore, we construct a fundamental set of solutions for the second order linear homogeneous βdifference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we derive the EulerCauchy βdifference equation.
3.1 Existence and uniqueness of solutions
Theorem 3.1
Let \(f_{i}(t,y_{1},y_{2}):I \times\prod_{i=1}^{2} S_{i}(x_{i}, b_{i})\rightarrow{\mathbb{X}}\), \(s_{0}\in I\), such that the following conditions are satisfied:

(i)
for \(y_{i}\in S_{i}(x_{i},b_{i})\), \(i=1,2\), \(f_{i}(t,y_{1},y_{2})\) are continuous at \(t=s_{0}\),

(ii)
there is a positive constant A such that, for \(t\in I\), \(y_{i}, \tilde{y}_{i}\in S_{i}(x_{i},b_{i})\), \(i=1,2\), the following Lipschitz condition is satisfied:
Then there exists a unique solution of the βinitial value problem, βIVP,
Proof
Let \(y_{0}=(x_{1},x_{2})^{T}\) and \(b=(b_{1},b_{2})^{T}\), where \((\cdot ,\cdot)^{T}\) stands for vector transpose. Define the function \(f:I\times \prod_{i=1}^{2}S_{i}(x_{i},b_{i})\rightarrow{\mathbb{X}}\times{\mathbb{X}}\) by \(f(t,y_{1},y_{2})= (f_{1}(t,y_{1},y_{2}),f_{2}(t, y_{1}, y_{2}) )^{T}\). It is easy to show that system (3.1) is equivalent to the βIVP
Since each \(f_{i}\) is continuous at \(t=s_{0}\), f is continuous at \(t=s_{0}\). The function f satisfies the Lipschitz condition because for \(y,\tilde {y}\in\prod_{i=1}^{2}S_{i}(x_{i},b_{i})\),
Applying Theorem 2.13, see the proof in [19], there exists \(\delta>0\) such that (3.2) has a unique solution on \([s_{0},s_{0}+\delta]\). Hence, the βIVP (3.1) has a unique solution on \([s_{0},s_{0}+\delta]\). □
Corollary 3.2
Let \(f(t,y_{1},y_{2})\) be a function defined on \(I\times\prod_{i=1}^{2} S_{i}(x_{i},b_{i})\) such that the following conditions are satisfied:

(i)
for any values of \(y_{i}\in S_{i}(x_{i},b_{i})\), \(i=1,2\), f is continuous at \(t=s_{0}\),

(ii)
f satisfies the Lipschitz condition
$$\big\ f(t,y_{1},y_{2})f(t,\tilde{y}_{1}, \tilde{y}_{2}) \big\ \leq A\sum_{i=1}^{2} \y_{i} \tilde{y}_{i}\, $$where \(A>0\), \(y_{i},\tilde{y}_{i}\in S_{i}(x_{i},b_{i})\), \(i=1,2\) and \(t \in I\). Then
$$\begin{aligned} D_{\beta}^{2}y(t)=f\bigl(t,y(t),D_{\beta}y(t)\bigr),\quad D_{\beta}^{i1}y(s_{0})=x_{i}, i=1,2 \end{aligned}$$(3.3)has a unique solution on \([s_{0},s_{0} +\delta]\).
Proof
Consider equation (3.3). It is equivalent to (3.1), where \(\{\phi_{i}(t)\}_{i=1}^{2}\) is a solution of (3.1) if and only if \(\phi_{1}(t)\) is a solution of (3.3). Here,
Hence, by Theorem 3.1, there exists \(\delta>0\) such that system (3.1) has a unique solution on \([s_{0},s_{0}+\delta]\). □
The following corollary gives us the sufficient conditions for the existence and uniqueness of the solutions of the βCauchy problem (3.3).
Corollary 3.3
Assume the functions \(a_{j}(t):I\rightarrow \mathbb{C}\), \(j=0,1,2\), and \(b(t):I\rightarrow{\mathbb{X}}\) satisfy the following conditions:

(i)
\(a_{j}(t), j=0,1,2\) and \(b(t)\) are continuous at \(s_{0}\) with \(a_{0}(t)\neq0\) for all \(t \in I\),

(ii)
\(a_{j}(t)/a_{0}(t)\) is bounded on I, \(j=1,2\). Then
$$ \begin{gathered} a_{0}(t)D_{\beta}^{2}y(t)+ a_{1}(t)D_{\beta}y(t)+a_{2}(t)y(t)=b(t), \\ D_{\beta}^{i1}y(s_{0})= x_{i},\quad x_{i} \in{\mathbb{X}}, i=1,2, \end{gathered} $$(3.4)has a unique solution on subinterval \(J\subseteq I\), \(s_{0}\in J\).
Proof
Dividing by \(a_{0}(t)\), we get
where \(A_{j}(t)=a_{j}(t)/a_{0}(t)\) and \(B(t)=b(t)/a_{0}(t)\). Since \(A_{j}(t)\) and \(B(t)\) are continuous at \(t=s_{0}\), the function \(f(t,y_{1},y_{2})\), defined by
is continuous at \(t=s_{0}\). Furthermore, \(A_{j}(t)\) is bounded on I. Consequently, there is \(A>0\) such that \(A_{j}(t)\leq A\) for all \(t \in I\). We can see that f satisfies the Lipschitz condition with Lipschitz constant A. Thus, \(f(t,y_{1},y_{2})\) satisfies the conditions of Corollary 3.2. Hence, there exists a unique solution of (3.5) on J. □
3.2 Fundamental solutions of linear homogeneous βdifference equations
The second order homogeneous linear βdifference equation has the form
where the coefficients \(a_{0}(t)\neq0\), \(a_{j}(t)\), \(j=1,2\) are assumed to satisfy the conditions of Corollary 3.3.
Lemma 3.4
If the function y is a solution of the homogeneous equation (3.6), such that \(y(s_{0})=0\) and \(D_{\beta}y(s_{0})=0\), \(s_{0}\in I\), then \(y(t)=0\), for all \(t\in J\).
Proof
By Corollary 3.3, if \(x_{i}=0\), \(i=1,2\) in the βIVP (3.4), which has a unique solution on J, then y such that \(y(t)=0\) for all \(t \in J\) is a unique solution of the βdifference equation (3.6), which satisfies the given initial conditions \(y(s_{0})=0\), \(D_{\beta}y(s_{0})= 0\). Hence we have the desired result. □
Theorem 3.5
The linear combination \(c_{1}y_{1}+c_{2}y_{2}\) of any two solutions \(y_{1}\) and \(y_{2}\) of the homogeneous linear βdifference equation (3.6) is also a solution of it in J, where \(c_{1}\) and \(c_{2}\) are arbitrary constants.
Proof
The proof is straightforward. □
Theorem 3.6
Let \(y_{1}\) and \(y_{2}\) be any two linearly independent solutions of the βdifference equation (3.6) in J. Then every solution y of (3.6) can be expressed as a linear combination \(y=c_{1}y_{1}+c_{2}y_{2}\).
Proof
Let
be the solutions of the linear system \(D_{\beta}y_{i}(t)=a_{i}(t)y_{i}(t)\), \(i=1,2\), corresponding, respectively, to the solutions \(y_{1}\), \(y_{2}\) of homogeneous linear βdifference equation (3.6). Since \(y_{1},y_{2}\) are linearly independent in J, then \(\phi_{1}\), \(\phi_{2}\) are linearly independent in J. Then there exist two constants \(c_{1}\), \(c_{2}\) such that \(\phi=c_{1}\phi _{1}+c_{2}\phi_{2}\). The first component of this is \(y=c_{1}y_{1}+c_{2}y_{2}\). Thus the results hold. □
Definition 3.7
A set of two linearly independent solutions of the second order homogeneous linear βdifference equation (3.6) is called a fundamental set of it.
Theorem 3.8
There exists a fundamental set of solutions of the second order homogeneous linear βdifference equation (3.6).
Proof
By Corollary 3.3, there exist unique solutions \(y_{1}\) and \(y_{2}\) of equation (3.6), such that \(y_{1}(s_{0})=1\), \(D_{\beta}y_{1}(s_{0})=0\) and \(y_{2}(s_{0})=0\), \(D_{\beta}y_{2}(s_{0})=1\).
Suppose that \(y_{1}\) and \(y_{2}\) are linear dependent, so there exist constants \(c_{1}\) and \(c_{2}\) not both zero, such that
We have \(c_{1}=c_{2}=0\) at \(t=s_{0}\), which is a contradiction. Thus the solutions \(y_{1}\) and \(y_{2}\) are linearly independent in J. Then there exists a fundamental set of the two solutions \(y_{1}\) and \(y_{2}\) of equation (3.6). □
Definition 3.9
Let \(y_{1}\), \(y_{2}\) be βdifferentiable functions. Then we define the βWronskian of the functions \(y_{1} \), \(y_{2}\), defined on I, by
Lemma 3.10
Let \(y_{1}(t)\), \(y_{2}(t)\) be functions defined on I. Then, for any \(t\in I\), \(t \neq s_{0}\),
Proof
Since \(W_{\beta}(y_{1},y_{2})(t)=y_{1}(t)D_{\beta}y_{2}(t)y_{2}(t)D_{\beta}y_{1}(t)\), then
which is the desired result. □
Theorem 3.11
Assume that \(y_{1}(t)\) and \(y_{2}(t)\) are two solutions of equation (3.6). Then their βWronskian, \(W_{\beta}\),
where \(r_{1}(t)=\frac{a_{1}(t)}{a_{0}(t)}\) and \(r_{2}(t)=\frac{a_{2} (t)}{a_{0}(t)}\) satisfy the conditions of Corollary 3.3.
Proof
Since \(y_{1}\) and \(y_{2}\) are solutions of equation (3.6), from (3.7) we have
which has the solution
□
Using Theorem 3.11 and Lemma 3.4, we can prove the following corollaries.
Corollary 3.12
Two solutions \(y_{1}\) and \(y_{2}\) of βdifference equation (3.6) are linearly dependent in J if and only if \(W_{\beta}(y_{1},y_{2})(t)=0\), for all \(t\in J\).
Corollary 3.13
The value of \(W_{\beta}(y_{1},y_{2})(t)\) of βdifference equation (3.6) either is zero or unequal to zero for all \(t \in J\).
3.3 Homogeneous equations with constant coefficients
Equation (3.6) can be written as
where a, b, and c are constants. The characteristic polynomial of equation (3.8) is
where \(y(t)=e_{\lambda,\beta}(t)\) is a solution of equation (3.8). Since equation (3.9) is a quadratic equation with real coefficients, it has two roots, which may be real and different, real but repeated, or complex conjugates.
Case 1: real and different roots of the characteristic equation ( 3.9 ).
Let \(\lambda_{1}\) and \(\lambda_{2}\) be real roots with \(\lambda_{1}\neq \lambda_{2}\), then \(y_{1}(t)=e_{\lambda_{1},\beta}(t)\) and \(y_{2}(t)=e_{\lambda_{2},\beta }(t)\) are two solutions of equation (3.8). Therefore,
is a general solution of equation (3.8), with
Example 3.14
Find the solution of the βinitial value problem
By assuming that \(y(t)=e_{\lambda,\beta}(t)\), we obtain the solution
Case 2: complex roots of the characteristic equation ( 3.9 ).
Let \(\lambda_{1}=\nu+i\mu\) and \(\lambda_{2}=\nui\mu\), where ν and μ are real numbers. Then \(y_{1}(t)=e_{(\nu+i\mu),\beta}(t)\) and \(y_{2}(t)=e_{(\nui\mu),\beta}(t)\) are two solutions of equation (3.8). By Theorems 2.10, 2.12, \(e_{(\nu+i\mu),\beta }(t)=e_{\nu,\beta}(t) e_{\frac{i\mu}{1+\nu(\beta(t)t)},\beta}(t)\). So,
We have
and
Therefore,
are two solutions of equation (3.8). If the βWronskian of u and v is not zero, then u and v form a fundamental set of solutions. The general solution of equation (3.8) is
where \(c_{1}\) and \(c_{2}\) are arbitrary constants.
Example 3.15
Find the general solution of
The characteristic equation is \(\lambda^{2}+\lambda+1=0\), and its roots are
Thus, the general solution of equation (3.10) is
Case 3: repeated roots.
Consider the case that the two roots \(\lambda_{1}\) and \(\lambda_{2}\) are equal, so
Therefore, the solution \(y_{1}(t)=e_{b/2a,\beta} (t)\) is one solution of the βdifference equation (3.8), and we give the second solution by the following example:
Example 3.16
Solve the βdifference equation
The characteristic equation is \((\lambda+2)^{2}=0\), so \(\lambda_{1}=\lambda _{2}=2\). Therefore, \(y_{1}(t)= e_{2,\beta}(t)\) is a solution of equation (3.11). To find the second solution, let \(y(t)=v(t)e_{2,\beta}(t)\). Then \({D_{\beta}^{2}v(t)=0}\). Therefore, \(v(t)=c_{1}t+c_{2}\), where \(c_{1}\) and \(c_{2}\) are arbitrary constants. Then the general solution is
where the two solutions \(y_{1}(t)=e_{2,\beta}(t)\) and \(y_{2}(t)=te_{2,\beta}(t)\) form a fundamental set of solutions of equation (3.11).
3.4 EulerCauchy βdifference equation
The EulerCauchy βdifference equation takes the form
where a, b are constants. The characteristic equation of (3.12) is given by
Theorem 3.17
If the characteristic equation (3.13) has two distinct roots \(\lambda_{1}\) and \(\lambda_{2} \), then a fundamental set of solutions of (3.12) is given by \(e_{\lambda_{1}/t,\beta}(t)\) and \(e_{\lambda _{2}/t,\beta}(t)\).
Proof
Let \(y(t)=e_{\lambda/t,\beta}(t)\), where λ is a root of equation (3.13). It follows that
Consequently, we have
Assume that \(\lambda_{1}\) and \(\lambda_{2}\) are distinct roots of the characteristic equation (3.13). Then, we have
Moreover, \(W_{\beta}(e_{\lambda_{1}/t,\beta},e_{\lambda_{2}/t,\beta})(t)\neq 0\), since \(\lambda_{1}\neq\lambda_{2}\). Hence, \(e_{\lambda_{1}/t,\beta}(t)\) and \(e_{\lambda_{2}/t,\beta}(t)\) form a fundamental set of solutions of (3.12). □
The following theorem gives us the general solution of the EulerCauchy βdifference equation in the double root case.
Theorem 3.18
Assume that \(1/\beta(t)\) is bounded on I and \(0\notin I\). Then the general solution of the EulerCauchy βdifference equation
is given by
Proof
The characteristic equation of (3.14) is
Then the characteristic roots are \(\lambda_{1}=\lambda_{2}=\gamma\). Hence one linearly independent solution of equation (3.14) is \(y_{1}(t)=e_{\frac{\gamma}{t},\beta}(t)\). To obtain the second linearly independent solution, we can rewrite equation (3.14) in the form
where \(r_{1}(t)=\frac{12\gamma}{\beta(t)}\) and \(r_{2}(t)=\frac{\gamma ^{2}}{t\beta(t)}\). Consequently,
Let u be a solution of equation (3.15) such that \(u(s_{0})=0\), \(D_{\beta}u(s_{0})=1\). Then
By Theorem 2.5, we find that u satisfies the following βdifference equation:
Then
Also,
Therefore,
is the general solution of equation (3.14). □
4 Conclusion
In this paper, the existence and uniqueness of solutions of the βCauchy problem of second order βdifference equations were proved. Moreover, a fundamental set of solutions for second order linear homogeneous βdifference equations when the coefficients are constants was constructed. Also, the different cases of the roots of the characteristic equations of these equations were studied. Finally, the EulerCauchy βdifference equation was derived.
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Faried, N., Shehata, E.M. & El Zafarani, R.M. On homogeneous second order linear general quantum difference equations. J Inequal Appl 2017, 198 (2017). https://doi.org/10.1186/s1366001714713
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DOI: https://doi.org/10.1186/s1366001714713