1. Introduction

Recently, many authors have used Laplace transform methods extensively for solving linear systems, see (^{[7, 10]}). For example, Laplace transform methods are used as a system by means of some differential equations with constant coefficients. However, some systems in engineering may be not both continuous and containing differential equations involving in computer controlled systems, telephone systems etc. Such systems are called sample systems which are degradation from a continuous signal to a discrete signal. The sample systems are related to difference equations. Using technique for solving sample systems is called Z-transform. The big role played by the Z-transform in the solution of sample systems corresponds to that played by the Laplace transforms in the solution of continuous systems.

Z-transform method is a fundamental way for sample systems. By means of sample systems contain difference equation, it is used to solve those difference equations. Note that zeros and poles and plotting in z-plane can be found by Z-transform (^{[11, 14]}).

Recently, q-calculus have been studying by many researchers who construct with different areas like mathematics, physics, medicine, statistic etc.. Many important notations in ordinary calculus were extended to q-calculus. q-analogue of special functions, polynomials and their many properties were obtained (^{[1, 2, 3]}). Also, q-analogue of some special matrices are extended with aid of Z-transform in (^[15]). A generalization of q-calculus is post-quantum calculus, denoted by (p,q)-calculus. This field have been studying by many researchers in approximation theory, analytic number theory, operator theory and so on. For example, (p,q)-Gamma and Beta functions, (p,q)-derivative, (p,q)-integration, (p,q)-Bernoulli, Euler and Genocchi polinomials were obtained (^{[4, 5, 12, 13]}).

This article is organised as follows:

In section 2, we give some basic results and notations that will be purposive in the conclusions.

In section 3, we present q and (p,q)-analogue of some elementary functions, numbers and Binom coefficients.

In final section, we present conclusions.

2. Miscellanous Definitions and Notations

The following definitions and notations enables us to obtain results in sequel.

Definition 1. Let be a sequence. The Z-transform of this sequence is defined as follows:

(2.1)

where z and above series are, respectively, a complex number and power series.

Definition 2. Let and be two Z-transform. Then, we have properties as follows:

(i)

(ii)

(iii)

(iv)

(v)

According to definition of Z- transform, of course, the Z-transform does not convergence for all sequences or for all values of z. A Z- transform have a set of values of z that called range of convergence (ROC). ROC have many properties for Z-transform. For example;

1) The ROC can not contain any poles.

2) The ROC is a ring or disk in the z-plane.

3) The ROC is a connected region.

Readers can look references (^[8]) for more knowledge about the Z-transform.

Definition 3. (^[13])The (p, q)-number that is called twin basic number is defined

(2.2)

where Note that in the case when it becomes

Definition 4. (^[13]) The (p, q)-Binom coefficients are defined as follows

(2.3)

(2.3) is a natural extension of the q-binomial coefficient.

Definition 5. (^[13]) The (p, q)-derivative operator is determined as (2.4)

(2.4)

where and

We give two type (p, q)-exponential functions like q-calculus as below:

Definition 6. (^{[6, 9]}) The (p, q)-exponential functions, seriatim, and are defined

(2.5)

(2.6)

For p = 1; we get q-exponential functions as follows:

(2.7)

(2.8)

By aid of q and (p, q)-exponential functions, q and (p, q)-trigonometric functions are determined.

Definition 7. (^{[6, 9]}) The q and (p, q)-trigonometric functions are

(2.9)

and

(2.10)

In the following section, we give some new results for numbers, notations and elementary functions under q- and (p, q)- calculus by using Z-transform.

3. Main Results

We are now in a position to start with the following Corollary 1.

Corollary 1. For we have Z-transform of (p, q)-number and (p, q)-Binomial coefficients as below:

(3.1)

and

(3.2)

where n is a natural number with the condition

Proof. Firstly, we will start with proof of (3.1). From def. (2.1) and some elementary operations

Now, we can prove (3.2) by the using definition of (p, q)-derivative:

where we use some properties of (p, q)-calculus [^[13]]:

We now consider Z-transform of (p, q)-exponential functions by the following corollary.

Corollary 2. For Z-transformers of (p, q)-exponential functions are obtained as below:

(3.3)

Proof.

where and are pole and root, seriatim, of Proof of Z-transform of can show with smilar way as above. In (3.3), we can obtain two types Z-transform of q-exponential functions when p = 1 as

We now state Z-transformers of (p, q)-trigonometric functions by using above corollary.

Corollary 3. Z-transformers of (p, q)-trigonometric functions are as follows:

(3.4)

Proof. If we use (2.10),

after some basic operations we get:

Z-transform of can obtain with similar way. Here, if we take p =1, (3.4) is reduced to q-sin e and q-cos ine functions as follows:

All of these new and old results are shown with a table:

Table

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Finally, we determine Z-transform of a (p, q)-array.

Lemma 1. For

(3.5)

Proof. We use mathematical induction for proof.

For we should show equality as below:

(3.6)

If we use convulation property, from (3.6)

Suppose that (3.5) holds true for We show that (3.5) is valid for

Therefore, the desired result is obtained.

4. Conclusions

In the paper, we give some results for q- and (p,q)- analogue of some numbers, notations and functions with concerned Z-transform. This results might useful likely for solving q-difference equations by means of Z-transform. In the next paper, we plan to deal with constructing (p,q)-difference equations to use the results obtained here. Also, we will deal with pascal matrix to extend in (p,q)-calculus with the help of Lemma 1.

Acknowledgement

Authors are thankful to the referees for their valuable comments that improved the presentation of this manuscript. The first author is also thankful to The Scientific and Technological Research Council of Turkey (TUBITAK) for their support with the Ph.D. scholarship.

References

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