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Some identities on r-central factorial numbers and r-central Bell polynomials
Advances in Difference Equations volume 2019, Article number: 245 (2019)
Abstract
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.
1 Introduction
For \(n\in \mathbb{N}\cup \{ 0\}\), as is well known, the central factorials \(x^{[n]}\) are defined by
It is also well known that the central factorial numbers of the second kind \(T(n,k)\) are defined by
where n is a nonnegative integer.
From (2), we can derive the generating function for \(T(n,k)\) (\(0 \leq k\leq n\)) as follows:
Recently, Kim and Kim [10] considered the central Bell polynomials given by
When \(x=1\), \(B_{n}^{(c)}=B_{n}^{(c)}(1)\) are called the central Bell numbers.
From (4), we can find the Dobinski-like formula for \(B_{n}^{(c)}(x)\):
The Stirling numbers of the second kind are defined by
where \((x)_{0}=1\), \((x)_{n}=x(x-1)(x-2)\cdots (x-n+1)\) (\(n\geq 1\)).
From (6), we can easily derive the following equation (7):
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 kind; 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 [1, 5, 7, 8, 10, 12]).
2 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 1
In [11], 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.
Theorem 1
For \(n,k,r\in \mathbb{N}\cup \{0\}\) with \(n\geq k\), we have
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.
Theorem 2
For \(n\geq 0\), we have
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
Remark 2
In [11], 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
By the comparison of the coefficients on both sides of (15), we can establish the following theorem.
Theorem 3
For \(n\geq 0\), we have that
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 4
For \(n,k\geq 0\), we have
By combining Theorems 3 and 4, we easily get
From (14), we have
Therefore, by comparing the coefficients on both sides of (22), we get the following identity.
Theorem 5
For \(n\geq 0\), we have
By (14), it can be checked that
Therefore, by comparing the coefficients on both sides of (23), we establish the following theorem.
Theorem 6
For \(n\geq 0\), we have
Now, we observe that
On the other hand, it can be seen that
Therefore, by (24) and (25), we obtain the following theorem.
Theorem 7
For \(m,n,k\geq 0\) with \(n\geq m+k\), we have
It is known that the generating function of central factorial is given by
If we let \(f(t)=2 \log (\frac{t}{2}+\sqrt{1+\frac{t^{2}}{4}} )\), 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
Therefore, by (28) and (29), the following identity is obtained.
Theorem 8
For \(n\geq 0\), we have the following identity:
3 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
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
Then we want to derive the generating function of the extended r-central factorial numbers of the first kind.
In addition, we also have
Now, from (40) and (41), we have the generating function for \(t_{r} (n+r,k+r )\):
Finally, we want to show a recurrence relation for the extended r-central factorial numbers of the first kind.
This verifies the following theorem.
Theorem 9
For any integers n, k with \(n-1 \geq k \geq 0\), we have the following recurrence relation:
4 Conclusions and discussion
In recent years, quite a number of old and new special numbers and polynomials have attracted attention of many researchers and have 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 kind; 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 [7, 8, 10, 12]).
As one of our next project, we would like to find some interesting applications of the numbers and polynomials introduced in this paper.
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This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1C1C1003869).
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Each of the authors, DSK, DVD, DK, and TK, contributed to each part of this study equally and read and approved the final version of the manuscript.
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Kim, D.S., Dolgy, D.V., Kim, D. et al. Some identities on r-central factorial numbers and r-central Bell polynomials. Adv Differ Equ 2019, 245 (2019). https://doi.org/10.1186/s13662-019-2195-0
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DOI: https://doi.org/10.1186/s13662-019-2195-0