1. Introduction
Let be a sequence of natural numbers, and be an array associated with this sequence, whose entries such that
The path of order k along is defined to be a sequence of entries as follows
The number of paths of order k will be denoted by
By neglecting the last row in and then reconsidering it, we get the recurrence
(1)
When, a is a constant, then
(2)
and
Khidr and El-Desouky [1] proved that, when
(3)
where are the generalized Stirling numbers of the first kind associated with the sequence of real numbers, defined by [1] - [6] ,
(4)
These numbers satisfy the recurrence relation
(5)
And
Moreover, they introduced a special case of (3), when, then the number of paths of order k, is denoted by; and proved that
(6)
where are the Stirling numbers of the first kind defined by, see [2] [3]
Also the generating function for is given by
(7)
In this article, in Section 2, we derive a generalization of some results given in [1] , for the number of paths of
order k, , when. The generating function of is given. In Section 3, we find the probability distribution for and study some of their properties. The moment
generating function, skewness and kurtosis for are investigated. Moreover special case and numerical results are given in Section 4.
2. Main Results
Theorem 1. The number of paths of order k is given by
(8)
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
(9)
Proof. Let the generating function of the number of paths of order k be denoted by
(10)
Using (1), we obtain
and hence we get
where. This completes the proof.
From (9), we get
where and hence we have
(11)
where
For the special case, we get
(12)
where
From (6) and (12), we have the identity
(13)
where
3. Some Applications
Let X, be the number of paths along, then by virtue of (8) we have
(14)
On the other hand the moment generating function of the random variable X denoted by, is given by the following theorem.
Theorem 3. The moment generating function of X, is given by
(15)
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
(16)
Proof. The jth moments can be obtained from the moment generating function, where
This completes the proof.
Then from (16), we can calculate the mean and variance for the random variable X as follows.
(17)
(18)
hence the variance is given by
(19)
Corollary 2. The Skewness and kurtosis for the random variable X are given by
(20)
where
Proof. We can find the jth moments about the mean by using
(21)
From (16) and (21), we can find the moments about mean which can be used to calculate the skweness and kurtosis.
Special Case:
If, from (14), we have
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, see [3] .
4. Numerical Results
Setting. Therefore the numerical values of, are reduced to, see [4] [5] .
From Equation (14), we can find the probability distribution of the number of paths X along as follows
From (16), we can compute the 4th moments as follows.
The 4th 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.