Complete and incomplete Bell polynomials associated with Lah-Bell numbers and polynomials

The nth r-extended 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 r-extended Lah-Bell polynomials and complete $r$-extended Lah-Bell polynomials respectively as multivariate versions of $r$-Lah numbers and the r-extended Lah-Bell numbers and to investigate some properties and identities for these polynomials. From these investigations, we obtain some expressions for the r-Lah numbers and the r-extended Lah-Bell numbers as finite sums.


INTRODUCTION
It is well known that the unsigned Lah number L(n, k), (n ≥ k ≥ 0), counts the number of ways a set with n elements can be partitioned into k non-empty linearly ordered subsets (see [4,7,8]). The n-th Lah-Bell number B L n , (n ≥ 0), is the number of ways a set with n elements can be partitioned into non-empty linearly ordered subsets. Thus, we note that (1) B L n = n ∑ k=0 L(n, k), (n ≥ 0), (see [7,8]).
Let n, k, r be non-negative integers with n ≥ k. Then, 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 (see [17]). The r-extended Lah-Bell number B L n,r 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 [7]). By the definitions of r-Lah numbers and r-extended Lah-Bell numbers, we have (4) B L n,r = n ∑ k=0 L r (n, k), (n ≥ 0), (see [7]).
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 , . . . , j n satisfying j 1 + 2 j 2 + · · · + n j n = n.
Thus, we note that where the sum runs over the set π(n, k) of all nonnegative integers ( j i ) i≥1 satisfying j 1 + j 2 + · · · + j n−k+1 = k, and 1 j 1 + 2 j 2 + · · · + (n − k + 1) j n−k+1 = n. Then the complete and incomplete Bell polynomials are related by Let f be a C ∞ -function. Then we have From (12) and (13), we obtain Kölbig and Coeffey's equation as follows : , (see [6,12]). The exponential incomplete r-Bell polynomials are defined by the generating function From (15), we note that where Λ(n, k, r) denotes the set of all non-negative integers (k i ) i≥1 and (r i ) i≥0 such that ∑ i≥1 k i = k, ∑ i≥0 r i = r and ∑ i≥1 i(k i + r i ) = n, (see [4,5,14]).
Assume that (a i ) i≥1 and (b i ) i≥1 are sequences of positive integers.
Then the number B (r) n+r,k+r (a 1 , a 2 , . . . ; b 1 , b 2 , . . . ) 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 r-Bell polynomials are given by (17) exp (see [4,5,11,14]). By (16) and (17), we get n+r,k+r (a 1 , a 2 , · · · : b 1 , b 2 , . . . ), (see [5]).
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 , · · · , x n−k+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 , · · · , x n−k+1 ) appear in the Faà di Bruno formula concerning higher-order derivatives of composite functions (see [5]). 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 Faà di Bruno algebra, etc (see [1]). The complete Bell polynomials B n (x 1 , x 2 , · · · , x n ) have applications to probability theory (see [5,12,18]). Indeed, the nth moment µ n = E[X n ] of the random variable X is the nth complete Bell polynomial in the first n cumulants µ n = B n (κ 1 , κ 2 , · · · , κ 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 number theoretic identities involving complete and incomplete Bell polynomials. We let the reader refer to the Introduction in [11] for further details on these.
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. We note here that the incomplete Bell polynomials (see (10)) and the incomplete Lah-Bell polynomials are related as given in (23), while the complete Bell polynomials (see (8)) and the complete Lah-Bell polynomials are related as given in (26). The incomplete r-extended Lah-Bell polynomials (see (30)) and the complete r-extended 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 r-Lah numbers and the r-extended Lah-Bell numbers.
The aim of this paper is to introduce the incomplete r-extended Lah-Bell polynomials and the complete r-extended Lah-Bell polynomials and to investigate some properties and identities for these polynomials. From these investigations, we obtain some expressions for the r-Lah numbers and the r-extended Lah-Bell numbers as finite sums. From (2), we note that Therefore, by (20) and (21), we obtain the following theorem.

CONCLUSION
There are various methods of studying special numbers and polynomials, for example, generating functions, combinatorial methods, umbral calculus, p-adic analysis, differential equations, probability theory, orthogonal polynomials, and special functions. These ways of investigating special polynomials and numbers can be applied also 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 making use of 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 number theoretic identities involving complete and incomplete Bell polynomials.
In this paper, we introduced the incomplete r-extended Lah-Bell polynomials and the complete r-extended Lah-Bell polynomials respectively as multivariate versions of r-Lah numbers and the rextended Lah-Bell numbers and investigated some properties and identities for these polynomials. As corollaries to these results, we obtained some expressions for the r-Lah 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 just as the incomplete and complete Bell polynomials have diverse applications.
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.