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Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials
Advances in Difference Equations volume 2021, Article number: 101 (2021)
Abstract
The nth rextended Lah–Bell number is defined as the number of ways a set with \(n+r\) elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete rextended Lah–Bell polynomials and complete rextended Lah–Bell polynomials respectively as multivariate versions of rLah numbers and the rextended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the rLah numbers and the rextended Lah–Bell numbers as finite sums.
1 Introduction
It is well known that the unsigned Lah number \(L(n,k)\) (\(n \ge k \ge 0\)) counts the number of ways a set with n elements can be partitioned into k nonempty linearly ordered subsets (see [4, 7, 8]). The nth Lah–Bell number \(B_{n}^{L}\) (\(n\ge 0\)) is the number of ways a set with n elements can be partitioned into nonempty linearly ordered subsets. Thus
From (1) it follows that the generating function of Lah–Bell numbers is given by
where
Explicitly, we see from (3) that the Lah numbers are given by
Let n, k, r be nonnegative integers with \(n\ge k\). Then the rLah 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 (see [17]). The rextended Lah–Bell number \(B_{n,r}^{L}\) is defined as the number of ways a set with \(n+r\) elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks (see [8]). By the definitions of rLah numbers and rextended Lah–Bell numbers we have
From (4) we see that the generating function of rextended LahBell numbers is given by
where
for nonnegative integers k.
Explicitly, the rLah numbers are given by
In [8] the rextended Lah–Bell polynomials are defined by
It is well known that the complete Bell polynomials are defined by
Then it can be shown that the complete Bell polynomials are given by
where the sum runs over all nonnegative integers \(j_{1},j_{2},\ldots ,j_{n}\) satisfying \(j_{1}+2j_{2}+\cdots +nj_{n}=n\). The incomplete Bell polynomials are given by
Thus
where the sum runs over the set \(\pi (n,k)\) of all nonnegative integers \((j_{i})_{i\ge 1}\) satisfying \(j_{1}+j_{2}+\cdots +j_{nk+1}=k\) and \(1 j_{1}+2 j_{2}+\cdots +(nk+1)j_{nk+1}=n\).
The complete and incomplete Bell polynomials are related by
Let f be a \(C^{\infty }\)function, that is, f is a function that has continuous derivatives of all orders on \((\infty ,\infty )\). Then by (8) we have
where \(f^{(j)}(x)\) is the jth derivative of \(f(x)\), and \(\exp (t)=e^{t}\).
We observe that
From (12) and (13) we obtain the Kölbig–Coeffey equation
The exponential incomplete rBell polynomials are defined by the generating function
From (15) we note that
where \(\Lambda (n,k,r)\) denotes the set of all nonnegative integers \((k_{i})_{i\ge 1}\) and \((r_{i})_{i\ge 0}\) such that
Let \((a_{i})_{i\ge 1}\) and \((b_{i})_{i\ge 1}\) are sequences of positive integers. Then the number \(B_{n+r,k+r}^{(r)}(a_{1},a_{2}, \ldots ;b_{1},b_{2},\ldots )\) counts the number of partitions of an \((n+r)\)set into \((k+r)\) blocks satisfying:

The first r elements belong to different blocks;

Any block of size i containing no elements from the first r elements can be colored with \(a_{i}\) colors;

Any block of size i containing one element from the first r elements can be colored with \(b_{i}\) colors.
The complete rBell polynomials are given by
The incomplete and complete Bell polynomials have applications to such diverse areas as combinatorics, probability, algebra, and analysis. The number of monomials appearing in the incomplete Bell polynomial \(B_{n,k}(x_{1},x_{2},\ldots ,x_{nk+1})\) is the number of partitioning n into k parts, and the coefficient of each monomial is the number of partitioning n as the corresponding k parts. Also, the incomplete Bell polynomials \(B_{n,k}(x_{1},x_{2},\ldots ,x_{nk+1})\) appear in the Faà di Bruno formula concerning higherorder derivatives of composite functions (see [6]). In addition, the incomplete Bell polynomials can be used in constructing sequences of binomial type (see [16]), and there are certain connections between incomplete Bell polynomials and combinatorial Hopf algebras such as the Hopf algebra of word symmetric functions, the Hopf algebra of symmetric functions, and the Fa di Bruno algebra (see [1]). The complete Bell polynomials \(B_{n}(x_{1},x_{2},\ldots ,x_{n})\) have applications to probability theory (see [6, 12, 18]). Indeed, the nth moment \(\mu _{n}=E[X^{n}]\) of the random variable X is the nth complete Bell polynomial in the first n cumulants \(\mu _{n}=B_{n}(\kappa _{1},\kappa _{2},\ldots ,\kappa _{n})\). The reader can refer to the Ph.D. thesis of Port [18] for many applications to probability theory and combinatorics. Many special numbers, like Stirling numbers of both kinds, Lah numbers, and idempotent numbers, appear in many combinatorial and numbertheoretic identities involving complete and incomplete Bell polynomials. We refer the reader to the Introduction in [11] for further details.
The incomplete Lah–Bell polynomials (see (22)) and the complete Lah–Bell polynomials (see (25)) are respectively multivariate versions of the unsigned Lah numbers and the Lah–Bell numbers. Note here that the incomplete Bell polynomials (see (10)) and the incomplete Lah–Bell polynomials are related as given in (23), whereas the complete Bell polynomials (see (8)) and the complete Lah–Bell polynomials are related as given in (26). The incomplete rextended Lah–Bell polynomials (see (30)) and the complete rextended Lah–Bell polynomials (see (32)) are respectively extended versions of the incomplete Lah–Bell polynomials and the complete Lah–Bell polynomials. Further, they are respectively multivariate versions of the rLah numbers and the rextended Lah–Bell numbers.
The aim of this paper is to introduce the incomplete rextended LahBell polynomials and the complete rextended LahBell polynomials and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the rLah numbers and the rextended Lah–Bell numbers as finite sums.
2 Complete and incomplete rextended Lah–Bell polynomials
Let \(f(t)=\frac{t}{1t}\). Then we have
By (14) we get
From (2) we note that
Therefore by (20) and (21) we obtain the following theorem.
Theorem 1
For \(n\ge 1\), we have
Let us consider the incomplete Lah–Bell polynomials given by
where \(n,k\ge 0\) with \(n\ge k\).
Note hat \(B_{n,k}^{L}(1,1,\ldots ,1)=L(n,k)\) (\(n \ge k \ge 0\)).
Indeed, by (10) and (22) we get
From (23) we note that
Therefore by (23) we obtain the following proposition.
Proposition 2
For \(n,k\ge 0\) with \(n\ge k\), we have
In addition,
From (23) we note that
We now consider the complete Lah–Bell polynomials given by
By (25) we get
From (22) and (25) we note that
Therefore by (25) and (27) we obtain the following theorem.
Theorem 3
For \(n\ge 1\), we have
In addition, for \(n\ge 1\), we have
where \(\pi (n,k)\) denotes the set of all nonnegative integers \((l_{i})_{i\ge 1}\) such that \(l_{1}+l_{2}+\cdots +l_{nk+1}=k\) and \(1\cdot l_{1}+2\cdot l_{2}+\cdots +(nk+1)l_{nk+1}=n\).
By (25) we easily get
From (28) we note that
By Proposition 2, (24), and Theorem 3 we get
Assume that \(\{a_{i}\}_{i\ge 1}\) and \(\{b_{i}\}_{i\ge 1}\) are sequences of positive integers. We define the incomplete rextended Lah–Bell polynomials by
where k, r are nonnegative integers.
From (30) we have
where \(\Lambda (n,k,2r)\) denotes the set of all nonnegative integers \(\{k_{i}\}_{i\ge 1}\) and \(\{r_{i}\}_{i\ge 0}\) such that \(\sum_{i\ge 1}k_{i}=k\), \(\sum_{i\ge 0}r_{i}=2r\), and \(\sum_{i\ge 1}i(k_{i}+r_{i})=n\).
We define the complete rextended Lah–Bell polynomials \(B_{n}^{(L,2r)}(x\mid a_{1},a_{2},\ldots :b_{1},b_{2},\ldots )\) (\(n\ge 0\)), which are given by
Thus we note that
By (18), (31), (32), and (34) we have
Therefore by (31) and (34) we obtain the following theorem.
Theorem 4
For \(n\ge 0\), we have
From (30) we note that
Therefore we obtain the following theorem.
Theorem 5
For \(n \ge k \ge 0\), we have
and
From (36) and (31) we note that
Corollary 6
For \(n,k,r\ge 0\) with \(n \ge k\), we have
where \(\Lambda (n,k,2r)\) denotes the set of all nonnegative integers \(\{k_{i}\}_{i\ge 1}\) and \(\{r_{i}\}_{i\ge }\) such that \(\sum_{i\ge 1}k_{i}=k\), \(\sum_{i\ge 0}r_{i}=2r\), and \(\sum_{i\ge 1}i(k_{i}+r_{i})=n\).
Now we observe that
and
Therefore by (32) and (41) we obtain the following theorem.
Theorem 7
For \(n,r\ge 0\), we have
Remark
For \(n\ge 0\), we have
Thus we note that
3 Conclusion
There are various methods of studying special numbers and polynomials, for example, generating functions, combinatorial methods, umbral calculus, padic analysis, differential equations, probability theory, orthogonal polynomials, and special functions. These ways of investigating special polynomials and numbers can be also applied to degenerate versions of such polynomials and numbers. Indeed, in recent years, many mathematicians have drawn their attention to studies of degenerate versions of many special polynomials and numbers by using the aforementioned means ([9, 10, 14] and references therein).
The incomplete and complete Bell polynomials arise in many different contexts as we stated in the Introduction. For instance, many special numbers, like Stirling numbers of both kinds, Lah numbers, and idempotent numbers, appear in many combinatorial and numbertheoretic identities involving complete and incomplete Bell polynomials.
In this paper, we introduced the incomplete rextended Lah–Bell polynomials and the complete rextended Lah–Bell polynomials respectively as multivariate versions of rLah numbers and the rextended Lah–Bell numbers and investigated some properties and identities for these polynomials. As corollaries of these results, we obtained some expressions for the rLah numbers and the rextended Lah–Bell numbers as finite sums.
It would be very interesting to explore many applications of the incomplete and complete rextended Lah–Bell polynomials as the incomplete and complete Bell polynomials have diverse applications.
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Acknowledgements
The authors would like to thank the reviewers for their valuable comments and suggestions and Jangjeon Research Institute for Mathematical Sciences for the support of this research.
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TK and DSK conceived of the framework and structured the whole paper; DSK and TK wrote the paper; LCJ, HL, and HYK checked the results of the paper; DSK and TK completed the revision of the paper. All authors have read and approved the final version of the manuscript.
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Kim, T., Kim, D.S., Jang, LC. et al. Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials. Adv Differ Equ 2021, 101 (2021). https://doi.org/10.1186/s13662021032583
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DOI: https://doi.org/10.1186/s13662021032583