Some identities on r-central factorial numbers and r-central Bell polynomials

In this paper, we introduce the extended r-central factorial numbers of the second and first kinds and the extended r-central Bell polynomials, as extended versions and central analogues of some previously introduced numbers and polynomials. Then we study various properties and identities related to these numbers and polynomials and also their connections.

It is also well known that the central factorial numbers of the second kind T (n, k) are defined by (2) x n = n ∑ k=0 T (n, k)x [k] , (see [4,5,16] where n is a nonnegative integer. From (2), we can derive the generating function for T (n, k), (0 ≤ k ≤ n) as follows: T (n, k) t n n! , (see [4,5,6]).
In this paper, we introduce the extended r-central factorial numbers of the second and first kinds and the extended r-central Bell polynomials, and study various properties and identities related to these numbers and polynomials and their connections. The extended r-central factorial numbers of the second kind are an extended version of the central factorial numbers of the second kind and also a central analogue of the r-Stirling numbers of the second; the extended r-central Bell polynomials are an extended version of the central Bell polynomials and also a central analogue of r-Bell polynomials; the extended r-central factorial numbers of the first kind are an extended version of the central factorial numbers of the first kind and a central analogue of the (unsigned) r-Stirling numbers of the first kind. All of these numbers and polynomials were studied before (see [4,5,8,14,15,18]).

EXTENDED r-CENTRAL FACTORIAL NUMBERS OF THE SECOND KIND AND EXTENDED
r-CENTRAL BELL POLYNOMIALS Let us first note that, by (3) and (4), Comparing the coefficients on both sides of (8), we have For any nonnegative integer r, we introduce the extended r-central factorial numbers of the second kind given by Remark 2.1. In [6], the extended central factorial numbers of the second kind were defined as Note that these numbers are different from the extended r-central factorial numbers of the second kind defined in (10).
From (3) and (10), we see that Therefore, by comparing the coefficients on both sides of (10), the following identity holds.
Next, we write e (r+x)t as follows: On the other hand, e (r+x)t can be written as Therefore, by the two expressions in (12) and (13) for e (r+x)t , we obtain the following identity.
In view of (4), we may now introduce the extended r-central Bell polynomials associated with the extended r-central factorial numbers of the second kind given by (14) e rt e Remark 2.4. In [6], the extended central Bell polynomials were defined as Observe here that these polynomials are different from the extended r-central Bell polynomials in (14).
From (14), we note that x k T r (n + r, k + r) t n n! .
By the comparison of the coefficients on both sides of (15), we can establish the following theorem. x k T r (n + r, k + r).
Next, we observe that By using the central difference operator δ , which is defined by we can show that We combine (18) with (16) to derive an equation for e (r+x)t as follows: From (10) and (19), we note that Therefore, by (20), we obtain the following theorem.
Theorem 2.6. For n, k ≥ 0, we have By combining Theorems 2.5 and 2.6, we easily get From (14), we have n−l (x)r l t n n! Therefore, by comparing the coefficients on both sides of (22), we get the following identity. By (14), it can be checked that Therefore, by comparing the coefficients on both sides of (23), we establish the following theorem. On the other hand, it can be seen that Therefore, by (24) and (25), we obtain the following theorem. It is known that the generating function of central factorial is given by x [n] t n n! , (see [4,5]).
If we let f (t) = 2 log t 2 + 1 + t 2 4 , then we can easily show that By the simple computations with the expressions in (1) and (2), we can check that e (x+r)t can be expressed as follows: Alternatively, the term e (x+r)t is also represented by (29) e (x+r)t = ∞ ∑ n=0 (x + r) n t n n! .
Theorem 2.10. For n ≥ 0, we have the following identity

EXTENDED r-CENTRAL FACTORIAL NUMBERS OF THE FIRST KIND
Throughout this section, we assume that r is any real number. The (unsigned) r-Stirling numbers of the first kind S 1,r (n + r, k + r) are defined by Then Further, we also have Combining (32) with (33), we obtain the generating function of S 1,r (n + r, k + r): The central factorial numbers of the first kind t (n, k) are defined by Using (26) and (35), we have On the other hand, we also have By combining (36) with (37), we get the generating function of t (n, k): Let us define the extended r-central factorial numbers of the first kind as Theorem 3.1. For any integers n, k with n − 1 ≥ k ≥ 0, we have the following recurrence relation:

CONCLUSIONS AND DISCUSSIONS
In recent years, quite a number of old and new special numbers and polynomials have attracted many researchers and been studied by means of generating functions, combinatorial methods, umbral calculus, differential equations, p-adic integrals, p-adic q-integrals, special functions, complex analysis and so on.
In this paper, we introduced the extended r-central factorial numbers of the second and first kinds and the extended r-central Bell polynomials, and studied various properties and identities related to these numbers and polynomials and their connections. This study was done by making use of generating function techniques.
The extended r-central factorial numbers of the second kind are an extended version of the central factorial numbers of the second kind and also a central analogue of the r-Stirling numbers of the second; the extended r-central Bell polynomials are an extended version of the central Bell polynomials and also a central analogue of r-Bell polynomials; the extended r-central factorial numbers of the first kind are an extended version of the central factorial numbers of the first kind and a central analogue of the (unsigned) r-Stirling numbers of the first kind. All of these numbers and polynomials were studied before (see [4,5,8,18]).
As one of our next project, we would like to find some interesting applications of the numbers and polynomials introduced in this paper.