Note on $r$-central Lah numbers and $r$-central Lah-Bell numbers

1.   Introduction

For $n \geq k \geq 0$, the Stirling numbers of the second kind $S_2(n, k)$ can be interpreted as the number of ways to partition a set with $n$ elements into $k$ non-empty subsets ^[7,19] and the $n$-th Bell number $B_n \; \; (n\geq 0)$ as the number of ways to partition a set with $n$ elements into non-empty subsets.

As is known, the $n$-th Bell polynomial possess the following explicit formula and generating function

respectively.

It is well-known that

where $< x > _n = x(x+1)(x+2) \cdots (x+n-1), \ \ (n\geq 1)$ and $< x > _0 = 1$.

The central factorial numbers of the second kind $T(n, k)$ are the coefficients in the expansion of $x^n$ in terms of central factorials given by

where $x^{[n]} = x\bigg(x+\frac{n}{2}-1\bigg)\bigg(x+\frac{n}{2}-2\bigg)\cdots \bigg(x-\frac{n}{2}+1\bigg), \ \ (n \geq 1)$ and $x^{[0]} = 1$. From (1.3), it is easy to see that the generating function of $T(n, k)$ is

where $k \geq 0$.

Throughout this section, let $n, k \in \mathbb{N}\cup \{0\}$ with $n\geq k\geq 0$ and $r \in \mathbb{N}$.

The $r$-Stirling numbers $S^{(r)}_2(n, k)$ of the second kind are defined by the generating function

and the $r$-Stirling numbers enumerate the numbers of partitions of the set $\{1, 2, \cdots, n\}$ into $k$ nonempty disjoint subsets in such a way that $1, 2, \cdots, r$ are in distinct subsets. The generating function of $r$-Bell polynomials $B^{(r)}_n(x) = \sum_{k = 1}^n S^{(r)}_2(n, k)$ is given by

In particular, when $x = 1$, $B^{(c, r)}_n = B^{(c, r)}_n(1)$, which are called the $r$-extended Bell numbers.

Kim et al. ^[10] introduced the $r$-extended central factorial numbers of the second kind given by

and the $r$-extended central Bell polynomials $B_n^{(c, r)}(x)$ given by

The unsigned Lah-number $L(n, k)$ counts the number of partitions of a set with $n$ elements into $k$ ordered blocks with no box left empty. Lah-numbers are rarely called Stirling numbers of the third kind. It is well known that the Lah number $L(n, k)$ respectively possess the following explicit formula and generating function

and

The $r$-Lah number $L_r(n, k)$ counts the number of partitions of a set with $n+r$ elements into $k+r$ ordered blocks such that $r$ distinguished elements have to be in distinct ordered blocks and $L_r(n, k)$ respectively possess the following explicit formula and generating function

and

Kim-Kim introduced the $r$-extended Lah-Bell polynomials given by the generating function

When $x = 1$, $\mathbf{B}_{n}^{L} = \mathbf{B}_{n}^{L}(1)$ and $\mathbf{B}_{r, n}^{L} = \mathbf{B}_{r, n}^{L}(1)$, which are called the Lah-Bell numbers and $r$-extended Lah-Bell numbers, respectively.

Recently, H. K. Kim introduced the central Lah numbers $L^{(C)}(n, k)$ and the central Lah-Bell numbers $LB_n ^{(C)}$ by means of the following generating functions

and

respectively. Where $LB^{(c)}_n = \sum_{k = 0}^{n}L^{(C)}(n, k)$.

Let $f$ be a non-negative real-valued function on interval $[a, b]$, and let $R = \{ (x, y) | a\leq x\leq b, \ \ 0 < y < f(x) \}$ be the region of the plane under $f(x)$ and above interval $[a, b]$. The Riemann integral

is approximations for the area of $R$, where $\triangle x_k = \frac{(b-a)k}{n}$ and $\triangle x_k^* = \frac{(b-a)k}{2n}$.

The central factorials have been introduced and studied by Steffensen ^[27]. The central factorial numbers are at least as important as Stirling numbers, and often appear in their properties and applications to difference calculus, spline theory, and to approximation theory, etc ^{[1,2,4,5,6,9,10,12,13,17,16]}. Moreover, the Lah numbers appear in various fields of mathematics such as non-crossing partitions, Dyck paths, q-analogues as well as falling and rising factorials ^{[7,8,10,13,14,16,18,20,21,23,25]}. Recently, H.K. Kim introduced and studied the central Lah numbers and the central Lah-Bell numbers. In this paper, we consider the $r$-central Lah numbers and the $r$-central Lah-Bell numbers ($r\in \mathbb{N}$), which generalize the central Lah numbers and the central Lah-Bell numbers. We derive the generating function, explicit formulas, binomial convolutions, and Riemann integral representations of the $r$-central Lah numbers and the $r$-central Lah-Bell numbers, respectively.

2.   $r$-central Lah numbers and $r$-central Lah-Bell numbers

In this section, we introduce the $r$-central Lah numbers and the $r$-central Lah-Bell numbers, respectively, which are "central" analogues for $r$-extended central factorial numbers of the second kind and $r$-extended central Bell polynomials. We investigate some properties of these numbers.

Now, in view of (1.5) and (1.11), we introduce the following generating function that defines the $r$-central Lah numbers $L_r^{(C)}(n, k)$

Theorem 1. For $n \geq k \geq 0$, an explicit formula of the $r$-central Lah-numbers $L_r^{(C)}(n, k)$ is

where $L(0, 0) = 1, \; \; L(n, 0) = 0$, and $L(n, k) = 0$ for all $k > n$.

Proof. We observe that from (1.2), (1.11) and (2.1),

By comparing the coefficients of both side of (2.2), we obtain the desired result.

Theorem 2. For $n \geq k \geq 0$, we have

where $L^{(C)}(n, k)$ are the central Lah-numbers.

Proof. From (1.2), (1.11) and (2.1), we observe that

By comparing the coefficients of both sides of (2.3), we get the desired result.

In view of (1.1), we define the $r$-central Lah-Bell polynomials $LB_{n, r}^{(C)}(x)$ by

when $x = 1$, $LB_{n, r}^{(C)}(1): = LB_{n, r}^{(C)}$ are the central $r$-Lah-Bell numbers.

Theorem 3. For $n \geq k \geq 0$, the generating function of the $r$-central Lah-Bell polynomials is

Proof. From (2.1) and (2.4), we have

By (2.5), we get the desired result.

Corollary 4. The generating function of the $r$-central Lah-Bell numbers $LB_{n, r}^{(C)}$ is

Lemma 5. For $n \geq k \geq 0$, we have

Proof. From (1.2) and (2.1), we observe that

By comparing the coefficients of both side of (2.6), we have desired result.

Theorem 6. For $n \geq k \geq 0$, we have

Proof. For $k \geq j \geq 0$,

By using Lemma 5, we obtain

From (2.7), we have desired result.

Theorem 7. For $n \geq 1$, we have

Proof. By (2.4), Theorem 3 and Theorem 6, we observe that

By (2.8), we have desired result.

Remark.

Corollary 8. For $n \geq k \geq 0$, we have

and

Proof. It is well known that the generating function of central factorial is given by

By replacing $t$ with $2\bigg(\frac{1}{2-t}-\frac{1}{2+t}\bigg)$ in (2.9), we get

Combining with (2.1), (2.4) and (2.10), we obtain

By comparing the coefficients of both sides of (2.11), we attain the desired result.

3.   Conclusions

In this paper, we introduced the $r$-central Lah numbers and the $r$-central Lah numbers $(r \in \mathbb{N})$, respectively. We derived the generating functions and combinatorial identities of the $r$-central Lah-Bell numbers and polynomials. Furthermore, by showing that each of these two numbers is expressed as Riemann integral in Theorems 6 and 7, respectively, we can infer approximate values for each of them. We also expressed the relation of the $r$-central Lah-numbers and the central factorials numbers in Corollary 8.

As one of our next projects, we would like to find some interesting applications of the $r$-central Lah numbers and polynomials, and the $r$-central Lah-Bell numbers and polynomials introduced in this paper.

Acknowledgments

The author thank Jangjeon Institute for Mathematical Science for the support of this research.

This work was supported by research grants from Daegu Catholic University in 2021.

Conflict of interest

The author declares no conflict of interest.