Some identities of Lah–Bell polynomials

Recently, the nth Lah–Bell number was defined as the number of ways a set of n elements can be partitioned into nonempty linearly ordered subsets for any nonnegative integer n. Further, as natural extensions of the Lah–Bell numbers, Lah–Bell polynomials are defined. We study Lah–Bell polynomials with and without the help of umbral calculus. Notably, we use three different formulas in order to express various known families of polynomials such as higher-order Bernoulli polynomials and poly-Bernoulli polynomials in terms of the Lah–Bell polynomials. In addition, we obtain several properties of Lah–Bell polynomials.


Introduction
The Stirling number of the second kind S 2 (n, k) is the number of ways to partition a set with n elements into k nonempty subsets. Thus B n = n k=0 S 2 (n, k), which is known as the nth Bell number, is the number of ways to partition a set with n elements into nonempty subsets. Further, the Bell polynomials B n (x) are natural extensions of the Bell numbers.
The Lah number L(n, k) counts the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets. So B L n = n k=0 L(n, k), which was recently defined as the nth Lah-Bell number (see [8]), counts the number of ways a set of n elements can be partitioned into nonempty linearly ordered subsets. In addition, the Lah-Bell polynomials B L n (x) are also defined as natural extensions of the Lah-Bell numbers. The aim of this paper is to study some properties of Lah-Bell polynomials with and without the help of umbral calculus. In particular, we represent several known families of polynomials in terms of the Lah-Bell polynomials, and vice versa. This has been done by using three different means, namely by using a formula derived from the definition of Sheffer polynomials (see Theorem 1), the transfer formula (see (29)), and the general formula expressing one Sheffer polynomial in terms of other Sheffer polynomial (see (12)). In more detail, we express Bernoulli polynomials, powers of x, poly-Bernoulli polynomials, and higher-order Bernoulli polynomials in terms of the Lah-Bell polynomials. In addition, we represent the Lah-Bell polynomials in terms of powers of x and of falling factorials. In addition, we obtain several properties of Lah-Bell polynomials. For the rest of this section, we recall some necessary facts that are needed throughout this paper and briefly review basic facts about umbral calculus. For more details on umbral calculus, we refer the reader to [13].
We recall from [8] that Lah-Bell polynomials B L n (x) are given by and the Lah-Bell numbers are given by B L n = B L n (1). For r ∈ N, the higher-order Bernoulli polynomials are given by t n n! (see [1,3,4,13]).
Let C be the field of complex numbers, and let F be the set of all power series in the variable t over C given by Let P = C[x], and let P * be the vector space of all linear functionals on P.

Some identities of Lah-Bell polynomials
Here we represent several known families of polynomials in terms of the Lah-Bell polynomials, and vice versa. This will be done by using three different means, namely by using a formula derived from the definition of Sheffer polynomials (see Theorem 1), the transfer formula (see (29)), and the general formula expressing one Sheffer polynomial in terms of other Sheffer polynomial (see (12)).
From (1) and (10), we note that where L(n, l) = n! l! n-1 l-1 are the Lah numbers given by Here the generating function of the Lah numbers in (15) can be easily derived either from power series expansion of the left-hand side of (15) or from the identity where x 0 = 1, x n = x(x + 1) · · · (x + n -1), n ≥ 1. Thus, we have For n ∈ N, by (18), we get Let Then P n is an (n + 1)-dimensional vector space over C. For P(x) ∈ P n , with P(x) = n m=0 A m B L m (x), we have By (20), we have Therefore, we obtain the following theorem.
From (2), we note that B n (x) = n l=0 n l B n-l x l ∈ P n .
For P(x) = B n (x) ∈ P n , we have where From (8), we easily note that (see [9,11,13]). Let us take P(x) = x n ∈ P n . Then, by Theorem 1, we have where Therefore, by (26) and (27), we obtain the following theorem.

Theorem 3
For n ≥ 0, we have For each nonnegative integer k, the differential operator t k on P is defined by Here (x) k is the falling factorial given by (x) 0 = 1, (x) k = x(x -1) · · · (xk + 1), k ≥ 1. Extending this linearly, any power series gives a differential operator on P defined by f (t)x n = n k=0 n k a k x n-k (n ≥ 0).
We consider the following two Sheffer sequences: From (28), (29), and (30), we note that Therefore, we obtain the following theorem. Let us consider the following two Sheffer sequences: and From (11) and (12), we note that Therefore, by (34) and (35), we obtain the following theorem.