1. Introduction
Riordan ( [1], pp. 213-217), defined the central factorial numbers of the first and second kind
and
, respectively
(1)
(2)
Equivalently, the
and
are determined by the recurrence relations
with
and
for
. If n and k are both odd, then
and
are not integers. For more details on the central factorial numbers, see Butzer et al. [2].
Kim et al. [3] extended
to the r-central factorial numbers of the second kind, r is a non-negative integer
In [4], the central factorial numbers with even arguments of both kinds are given by
(3)
and the central factorial numbers with odd arguments of both kinds are given by
(4)
Note that
and
are integers for all
. The combinatorial interpretations of these numbers can be found in [4] and the references therein.
Recently, Shiha [5] introduced the r-cental factorial numbers with even arguments of the first (resp. the second) kind
(resp.
), and introduced many properties and identities for these numbers. For all integers
,
(5)
(6)
In the next, we consider a polynomial generalization of the cental factorial numbers with odd arguments of the first and second kind, which we will denote by
and
, respectively. The distribution of the signless r-central factorial numbers with odd arguments of the first kind is derived. Moreover, we give many properties of these new numbers, including a new and interesting connection between these numbers and the Legendre-Stirling numbers.
2. The Generalized Central Factorial Numbers with Odd Arguments
Definition 1. Given integers
, the arrays
and
are defined by
(7)
and
(8)
In particular, if
, the numbers
are reduced to
and
are reduced to
.
These numbers satisfy the following orthogonality relation:
(9)
The numbers
and
satisfy the following two-term recurrence relations.
Theorem 1. The arrays
and
for
are satisfy the recurrence
(10)
and
(11)
with
,
and
for
.
Proof. From (7), we have
Equating the coefficients of
on both sides, we obtain Equation (10). For
, we find
Successive application gives
. The proof for (11) is similarly.
Moreover, we derive explicit formulas and further recurrences satisfied by
and
by using the following theorem.
Proposition 2. (Mansour et al. [6]) Suppose that the array
is defined by
(12)
with
and
, for all
, where
and
are given sequences with the
distinct, then
(13)
and
(14)
Theorem 3. For any integer
,
(15)
(16)
(17)
Proof. Setting
and
for all j in (13), then
Since
, then
For (16), set
,
in (14), and for (17), set
,
in (14).
To get the exponential generating function of
, multiply both sides of (15) by
and summing over
,
(18)
3. The Distribution of
The signless r-central factorial numbers of odd arguments of the first kind is defined as
Theorem 4. The array
has a Poisson-binomial distribution.
Proof. Define the random variables
,
, such that
(19)
The probability generating function of
is given by
(20)
Then
can be represented as a total number of successes in n independent Bernoulli trials where
is the probability of success at trial i. Thus, the random variable
has a Poisson-binomial distribution and hence, the array
.
4. Generating Function Formulas
In this section, we give the generating function formulas and some related identities for the numbers
and
.
Theorem 5. If
, then
(21)
(22)
Proof. Replacing x by
in (7), and multiplying both sides by
, gives
an hence replace z by
,
then replacing k by
gives (21). For (22), let
, hence the initial condition is given by
By virtue of (11),
hence
Iterating this recurrence, gives (22).
For a set of variables
, the k-th elementary symmetric function
and the k-th complete homogeneous symmetric function
are given, respectively, by
with
, and
for
or
.
The generating functions of
and
are given by, see [7]
(23)
(24)
Using (21) and (22), it is not difficult to show that
and
are the specializations of the elementary and complete symmetric functions, i.e.,
(25)
(26)
In particular, at
, the central factorial numbers with odd arguments of the first kind are the elementary symmetric functions of the numbers
, i.e.,
(27)
and the central factorial numbers with odd arguments of the second kind are the complete homogeneous symmetric functions of the numbers
, i.e.,
(28)
Theorem 6. (Merca [8]) Let k and n be two positive integers, then
(29)
and
(30)
where
are variables.
In the next theorem, we prove that the central factorial numbers with odd arguments can be expressed in terms of r-central factorial numbers with odd arguments and vice versa.
Theorem 7. For
, we have
1)
,
2)
,
3)
,
4)
.
Proof. By using (25) and Equation (29)
Replacing k by
,
gives the first identity. From (27) and (29), we get
By replacing k by
and then
by
, we get the second identity. The last two identities can be proven similarly by using the relations (26), (28) and (30).
5. The Generalized Central Factorial Matrices with Odd Arguments
Matrix representation and factorization for the special numbers are well developed by many authors, see for example [5] [8] [9] [10] [11]. In the following, we define the r-central factorial matrices with odd arguments of both kinds and give factorizations for them.
Definition 2. The r-central factorial matrix with odd arguments of the first kind is the
matrix defined by
Similarly, the r-central factorial matrix with odd arguments of the second kind is the
matrix defined by
When
, we obtain the central factorial matrices with odd arguments of both kinds,
For example,
and
The orthogonality property (9) gives the following identity
The generalized
Pascal matrix
(see [12]) is defined as:
(31)
with
, the Pascal matrix of order n. Moreover,
From Theorem 7, we have the important matrix representations
(32)
and
(33)
For example
and
6. The Generalized Central Factorial Numbers and Legender-Stirling Numbers
The Legendre-Stirling numbers were introduced by [13], and many properties of these numbers have been studied later in [14] [15].
The Legendre-Stirling numbers of the first kind
are defined by
and the Legendre-Stirling numbers of the second kind
are defined by
In fact, the Legendre-Stirling numbers are specializations of the elementary and complete homogeneous symmetric functions, i.e.,
(34)
(35)
We next give some connections between the r-central factorial numbers with odd arguments and the Legendre-Stirling numbers.
Theorem 8. For
,
(36)
(37)
(38)
(39)
Proof. For (36), we note that
Then
For (37), by virtue of (25),
Hence
The proofs of (38) and (39) are similar.
For example, for
, from (37) we have
and for (36),
For example, for
, from (39) we have
and for (38),
7. Conclusion
The r-central factorial numbers with odd arguments of both kinds are defined. We obtained recurrence relations, generating functions and explicit formulas of these numbers. Matrix representation and the relation between these numbers and Pascal matrix are given. The distribution of the signless r-central factorial numbers of odd arguments of the first kind is derived. Finally, connections between the r-central factorial numbers with odd arguments and the Legendre-Stirling numbers are investigated.