By using the basic (or

A function

Note that the notation

Furthermore, let us consider a subclass

It should be noted that for

For a parameter

The Poisson distribution is a statistical distribution that calculates the probability of a certain number of events occurring in a particular time period. The Poisson distribution is commonly used to model rate of random events that occur (arrive) in some fixed time interval. The Poisson distribution models packet arrival times as an independent identically distributed process with an exponential distribution. However, it has been demonstrated in reality that packet inter-arrival durations do not follow an exponential distribution, resulting in a considerable increase in the error caused by modeling them as a Poisson distribution. User-initiated TCP (Transmission Control Protocol) session arrivals, such as remote login and file transfer, are well-modeled as Poisson processes with fixed hourly rates, but other connection arrivals deviate significantly from Poisson; modeling TELNET (Teletype Network Protocol) packet inter arrivals as exponential greatly underestimates the burstiness of TELNET traffic, according to studies.

The power series

with its coefficients as probabilities of Poisson distribution which was introduced by Porwal [

More about special functions and related topics, we may refer to [

The Basic (or

In 1748 Euler studied a generating function for _{n}_{2}_{1} hypergeometric series.

Beside from the influential study of Rogers and Thomae the subject stayed moderately comatose throughout the latter part of the nineteenth century up till Jackson embarked on a long lasting program of developing the theory of basic hypergeometric series in an organized mode (see [

We next recollect certain elementary and useful concept details of the

_{q}

provided that

In a given subset of

In Geometric Function Theory of Complex Analysis, the role of the _{q}_{q}

More recently, Srivastava published a review article [_{q}

Motivated by the above-mentioned works of Srivastava [

In this section, making use of the concept of

For the following function

If the complex number

We see that the last inequality is equivalent to

Conversely, suppose that

On the other hand, we see that

It is now easy to see from

The equality holds for the function

We define the function

Furthermore, we now give certain closure results for the function involved in

Now making use of Theorem 2.1, we have

Now by Theorem 2.1, the proof of Theorem 2.3 is completed.

is in the class

By Theorem 2.1, we have

The Euler distribution or Heine distribution, as shown in [

Furthermore, the theory of

The graph of

In other words, if

In the following equations we settled a power series so that its coefficients are probabilities the of

It could be seen that by ratio test the radius of convergence of

According to Theorem 2.1, we must show that

Therefore, we now consider

The last expression in

Thus we complete the required proof of our Theorem.

According to Theorem 2.2, we must show that

Therefore, we now consider

We in last expression see that the

Thus we have completed the proof of our Theorem.

In our present work, by using the

The usage of basic (or