Received 11 October 2015; accepted 16 January 2016; published 19 January 2016
The Laplace transform provides an effective method for solving linear differential equations with constant coefficients and certain integral equations. Laplace transforms on time scales, which are intended to unify and to generalize the continuous and discrete cases, were initiated by Hilger  and then developed by Peterson and the authors  .
2. The q-Laplace Transform
Definition 2.1. A time scale T is an arbtrary nonempty closed subset of the real numbers. Thus the real numbers R, the integers Z, the natural numbers N, the nonnegative integers, and the q-numbers with fixed are examples of time scales   .
Definition 2.2. Assume is a function and. Then we define to be the number with the property that given any, there is a nighbourhood U (in T) of t such that
We call the delta (or Hilger) derivative of f at t.
is the usual Jakson derivative if.
Definition 2.3. If is a function, then its q-Laplace transform is defined by
for those values of, , for which this series converges, where.
Let us set
which is a polynomial in Z of degree. It is easily verified that the equations
hold, where. The numbers
where, belong to the real axis interval and tend to zero as. For any and, we set
so that is a closed domain of the complex plane C, whose points are in distance not less than from the set.
Lemma 2.4. For any,
Therefore, for an arbitrary number, there exists a positive integer such that
Example 2.5. We find the q-Laplace transform of (k is a fixed number). We have in,
Example 2.6. We find the q-Laplace transform of the functions and .
We have (see  ),
On the other hand, we know that
with respect to
The q-Laplace transform of the functions and, would be
Theorem 2.7. If the function satisfies the condition
where c and R are some positive constants, then the series in (1) converges uniformly with respect to z in the region and therefore its sum is an analytic (holomorphic) function in.
Proof. By Lemma 2.4, for the number R given in (8) we can choose an such that
Then for the general term of the series in (1), we have the estimate
Hence the proof is completed.
A larger class of functions for which the q-Laplace transform exists is the class of functions satisfying the condition
Theorem 2.8. For any, the series in (1) converges uniformly with respect to z in the region, and therefore its sum is an analytic function in.
Proof. By using the reverse (5), hence
and comparison test to get the desired result.
Theorem 2.9. (Initial Value and Final Value Theorem). We have the following:
a) If for some, then
b) If for all, then
Proof. Assume for some. It follows from (1) that
Multiplying, on both sides of the relation of (12) and by using equivalence relation, which yields (10). Note that we have taken a term-by-term limit due to the uniform convergence (Theorem 2.8) of the series in the region.
Definition 3.1. Let T be a time scale. We define the forward jump operator by
Definition 3.2. For a given function, its shift (or delay) is defined as the solution of the problem
Definition 3.3. For given functions, their convolution is defined by
where is the shift of f introduced in Definition 3.2  .
Definition 3.4. For given functions, their convolution is defined by
Theorem 3.5. (Convolution Theorem). Assume that, , and exist for a given. Then at the point z,
4. Concluding Remarks
1) We can see from Theorem 2.9(a) that no function has its q-Laplace transform equal to the constant function 1.
2) Finally, we note that most of the results concerning the Laplace transform on can be generalized appropriately to an arbitrary isolated time scale such that