Some identities for a sequence of unnamed polynomials connected with the Bell polynomials

In the paper, by virtue of (1) the Stirling inversion theorem and the binomial inversion theorem, (2) the Faà di Bruno formula and two identities for the Bell polynomials of the second kind, (3) a formula of higher order derivative for the ratio of two differentiable functions, the authors (1) present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, (2) derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, (3) recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.


Motivations
In [4, pp. 257-258] and [5], the function F(t, x) and the sequence of polynomials h n (x) defined by were considered. In [4, pp. 257-258], it was pointed out that the sequence of unnamed polynomials h n (x) satisfies (1.2) In [5], it was pointed out that the sequence of unnamed polynomials h n (x) and the Bell polynomials B n (x) are connected by the identity It was pointed out in [4, pp. 257-258] that the expression (1.2) had been applied in 1937 to the theory of hyperbolic differential equations. It was also pointed out in [17] that there have been some studies on interesting applications of the Bell polynomials B k (x) in soliton theory, including links with bilinear and trilinear forms of nonlinear differential equations which possess soliton solutions. See, for example, [6][7][8]. Therefore, applications of the Bell polynomials B k (x) to integrable nonlinear equations are greatly expected and any amendment on multi-linear forms of soliton equations, even on exact solutions, would be beneficial to interested audiences in the research community.
To simplify main results in [5], among other things, the following conclusions were established in the newly published paper [17].
where the double factorial of negative odd integers −(2n + 1) is defined by and the conventions 0 0 = 1 and p q = 0 for q > p ≥ 0 are adopted. Consequently, the sequence of polynomials h n (x) for n ≥ 0 can be expressed as where s(n, k) for n ≥ k ≥ 0, which can be generated by stand for the Stirling numbers of the first kind.
In this paper, making use of two inversion theorems for the Stirling numbers s(n, k) and S(n, k) and for binomial coefficient n k , employing the Faà di Bruno formula and two identities of the Bell polynomials of the second kind B(n, k), and utilizing a formula of higher order derivative for the ratio of two differentiable functions, we will present two explicit formulas, a determinantal expression, and a recursive relation for h n (x), derive two identities connecting h n (x) with B n (x), and recover the identity (1.3).
Our main results can be stated as the following four theorems.
is the falling factorial.

Theorem 1.5 For n ≥ 0, the Bell polynomials B n (x) and h n (x) satisfy
and by the explicit formula

Lemmas
In order to prove our main results, we recall several lemmas below.
The Faà di Bruno formula can be described in terms of the Bell polynomials of the second where a and b are any complex numbers.
for k ≥ 0. In other words, the formula (2.4) can be rewritten as

Proofs of main results
Now we are in a position to prove our main results.
The proof of Theorem 1.4 is complete.
Proof of Theorem 1.5 The identity (1.6) in Theorem 1.3 can be rearranged as Further utilizing Lemma 2.2 yields The proof of Theorem 1.5 is complete.
Proof of Theorem 1. 6 Interchanging the order of sums in the right hand side of the identity (1.4) in Theorem 1.2 arrives at Further applying Lemma 2.3 leads to The proof of Theorem 1.6 is complete.

Remark 4.5
In the recent papers [9,10,19], it was found that the Bell polynomials B k (x) can be applied to white noise distribution theory. Remark 4.7 This paper is a slightly revised version of the preprint [18].