^{1}

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^{2}

Khidr and El-Desouky [1] derived a symmetric sum involving the Stirling numbers of the first kind through the process of counting the number of paths along a rectangular array
*n**
*m* denoted by
* A*
_{nm}. We investigate the generating function for the general case and hence some special cases as well. The probability function of the number of paths along is obtained. Moreover, the moment generating function of the random variable X and hence the mean and variance are obtained. Finally, some applications are introduced.

Let

The path of order k along

The number of paths of order k will be denoted by

By neglecting the last row in

When

and

Khidr and El-Desouky [

where

These numbers satisfy the recurrence relation

And

Moreover, they introduced a special case of (3), when

where

Also the generating function for

In this article, in Section 2, we derive a generalization of some results given in [

order k,

generating function, skewness and kurtosis for

Theorem 1. The number of paths of order k is given by

Proof. Using (5) in (8), we get

This by virtue of (1) completes the proof of (8).

Theorem 2. The generating function of the number of paths of order k is given by

Proof. Let the generating function of the number of paths of order k be denoted by

Using (1), we obtain

and hence we get

where

From (9), we get

where

where

For the special case

where

From (6) and (12), we have the identity

where

Let X, be the number of paths along

On the other hand the moment generating function of the random variable X denoted by

Theorem 3. The moment generating function of X, is given by

Proof. We begin by the definition of the moment generating function as follows.

This completes the proof.

Corollary 1. The jth moments of X is

Proof. The jth moments can be obtained from the moment generating function,

This completes the proof.

Then from (16), we can calculate the mean and variance for the random variable X as follows.

hence the variance is given by

Corollary 2. The Skewness and kurtosis for the random variable X are given by

where

Proof. We can find the jth moments about the mean by using

From (16) and (21), we can find the moments

Special Case:

If

and from (16) the jth moments has the form

and the mean is given by

the variance can be obtained as follows.

where we used

Setting

From Equation (14), we can find the probability distribution of the number of paths X along

From (16), we can compute the 4th moments as follows.

The 4^{th} moments about mean can be obtained as

The values of mean and variance can be obtained from (17) and (19) as follows.

The skewness and kurtosis, respectively can be obtained from (20) as follows.