GENERALIZATIONS OF THE BELL NUMBERS AND POLYNOMIALS AND THEIR PROPERTIES

In the paper, the authors present unified generalizations for the Bell numbers and polynomials, establish explicit formulas and inversion formulas for these generalizations in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem connected with the Stirling numbers of the first and second kinds, construct determinantal and product inequalities for these generalizations with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations. E-mail addresses: qifeng618@gmail.com, qifeng618@hotmail.com, nnddww@gmail.com, dgrim84@gmail.com, bai.ni.guo@gmail.com, bai.ni.guo@hotmail.com. 2010 Mathematics Subject Classification. Primary 11B83; Secondary 11A25, 11B73, 11C08, 11C20, 15A15, 26A24, 26A48, 26C05, 26D05, 33B10, 34A05.


Motivations
In combinatorial mathematics, the Bell numbers B n for n ∈ {0} ∪ N, where N denotes the set of all positive integers, count the number of ways a set with n elements can be partitioned into disjoint and nonempty subsets.These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, a Scottishborn mathematician and science fiction writer who lived in the United States for most of his life and wrote about B n in the 1930s.The Bell numbers B n for n ≥ 0 can be generated by For detailed information on the Bell numbers B n , please refer to [2,4,5,6,16,17,21] and plenty of references therein.
In the paper [19] it was pointed out that there have been studies in [9,10,11] on interesting applications of the Bell polynomials B n (x) in soliton theory, including links with bilinear and trilinear forms of nonlinear differential equations which possess soliton solutions.Therefore, applications of the Bell polynomials B n (x) to integrable nonlinear equations are greatly expected and any amendment on multilinear forms of soliton equations, even on exact solutions, would be beneficial to interested audiences in the research community.For more information about the Bell polynomials B n (x), please refer to [7,19,20] and closely related references therein.
For n ≥ 2, let In [1], the quantities were called the Bell numbers of order n ≥ 2 and were deeply studied.In [1, Section 3] and [3,8], the Bell numbers b n (k) of order n were applied to white noise distribution theory.Since not all these numbers are integers, we can say that the Bell numbers b n (k) of order n are not good generalizations of the Bell numbers In this paper, we will present better and unified generalizations B m,n (x m ) for the Bell numbers B n and the Bell polynomials B n (x), establish explicit formulas and inversion formulas for these generalizations B m,n (x m ) in terms of the Stirling numbers of the first and second kinds s(n, k) and S(n, k) with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind B n,k (x 1 , x 2 , . . ., x n−k+1 ), and the inversion theorem connected with the Stirling numbers s(n, k) and S(n, k), construct determinantal and product inequalities for these generalizations B m,n (x m ) with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations B m,n (x m ).

Generalizations of the Bell numbers and polynomials
For m ∈ N, let In other words, the generating function f (t; 2) and, by virtue of the software Mathematica, (2.4) The equality (2.3) means that B m,0 (x 1 , x 2 , . . ., x m−1 , x m ) = 1 for all m ∈ N.
For conveniently referring to the quantities B m,n (x 1 , x 2 , . . ., x m−1 , x m ) for n ≥ 0, we need a name for them.We would like to recommend a name for them: the Bell polynomials of m variables x 1 , x 2 , . . ., x m−1 , x m .For the sake of simplicity and uniqueness, in what follows, we call them the Bell-Qi polynomials.
Let x m = (x 1 , x 2 , . . ., x m ).Occasionally for shortness in symbols, we use the notation for n ≥ 0 the Bell-Qi numbers or the Bell numbers of m multiples and denote them by B m,n .
It is not difficult to see that we can take t and x k for 1 ≤ k ≤ m in B m,n (x m ) on the complex plane C.

Explicit formulas for the Bell-Qi numbers and polynomials
Roughly judging from concrete expressions in (2.4), the Bell-Qi polynomial B m,n (x 1 , x 2 , . . ., x m−1 , x m ) for m ∈ N and n ≥ 0 should be a polynomial of m variables x 1 , x 2 , . . ., x m−1 , x m with degree m × n and positive integer coefficients.This guess will be verified by the following theorem.
where 0 = n and S(n, k) for n ≥ k ≥ 0, which can be generated by represent the Stirling numbers of the second kind.Consequently, the Bell-Qi numbers B m,n can be computed explicitly by Proof.In combinatorial analysis, the Bell polynomials of the second kind B n,k are defined by see [4, p. 135], where a and b are any complex numbers.The Faà di Bruno formula for computing higher order derivatives of composite functions can be described in terms of the Bell polynomials of the second kind B n,k by where u 0 = u 0 (t) = t and u q = u q (u q−1 ) = x m−q+1 (e uq−1 − 1), 1 ≤ q ≤ m.
When t → 0, it follows that u q → 0 for all 0 ≤ q ≤ m.As a result, by the definition in (2.1), we have The formula (3.1) is thus proved.The formula (3.2) follows from taking . The proof of Theorem 3.1 is complete.
Remark 3.1.When letting m = 1, 2, 3 in (3.1), we can recover and find explicit formulas and for n ≥ 0. The formula (3.6) was also recovered in [18,Theorem 3.1].The formulas (3.7) and (3.8) coincide with those special values in (2.4).This convinces us that Theorem 3.1 and its proof in this paper are correct.
Theorem 3.2.For m ∈ N and n ≥ 0, the Bell-Qi polynomials B m,n (x m ) satisfy m q=1 1 where 0 = n and s(n, k) for n ≥ k ≥ 0, which can be generated by stand for the Stirling numbers of the first kind.Consequently, the Bell-Qi numbers B m,n satisfy the identity Proof.The formula (3.1) can be rearranged as In [22, p. 171 and, inductively, which can be further rewritten as and the identity (3.9).The identity (3.10) follows from taking ).The proof of Theorem 3.2 is complete.Remark 3.2.When letting m = 1, 2, 3 in (3.9), we can recover and derive and for n ≥ 0. The identity (3.12) was also obtained in [ It is seemingly true that there have been more identities than inequalities in combinatorial mathematics.Now we start out to construct some determinantal and product inequalities for the Bell-Qi numbers B m,n and the Bell-Qi polynomials B m,n (x m ).Consequently, we can derive that the sequences of the Bell-Qi numbers B m,n and the Bell-Qi polynomials B m,n (x m ) are logarithmically convex.
Theorem 4.1.Let q ≥ 1 be a positive integer, let |e ij | q denote a determinant of order q with elements e ij , and let x k > 0 for 1 ≤ k ≤ m.
(1) If a i for 1 ≤ i ≤ q are non-negative integers, then and B m,ai+aj (x m ) q ≥ 0. ( (2) If a = (a 1 , a 2 , . . ., a q ) and b = (b 1 , b 2 , . . ., b q ) are non-increasing q-tuples of non-negative integers such that Proof.Recall from [14, Chapter XIII] and [26, Chapter IV] that a function f is said to be absolutely monotonic on an interval I if it has derivatives of all orders and f (k−1) (t) ≥ 0 for t ∈ I and k ∈ N. Recall from [14, Chapter XIII], [23,Chapter 1], and [26, Chapter IV] that an infinitely differentiable function f is said to be completely monotonic on an interval I if it satisfies (−1) k f (k) (x) ≥ 0 on I for all k ≥ 0. Theorem 2b in [26, p. 145] reads that, if f 1 (x) is absolutely monotonic and f 2 (x) is completely monotonic on their defined intervals, then their composite function f 1 (f 2 (x)) is completely monotonic on its defined interval.Therefore, since e t and e −t are respectively absolutely and completely monotonic on [0, ∞), by induction, it is not difficult to reveal that, when x 1 , x 2 , . . ., x m > 0, the generating function f (−t; x 1 , . . ., x m ) is completely monotonic with respect to t ∈ [0, ∞).Moreover, by (2.1), it is obvious that For simplicity, in what follows, we use f (±t, x m ) for f (±t; x 1 , . . ., x m ).

preprints.org) | NOT PEER-REVIEWED | Posted: 26 August 2017 doi:10.20944/preprints201708.0090.v1
, Theorem 12.1], it is stated that, if b α and a k are a collection of constants independent of n, then Applying this inversion theorem to(3.11)consecutivelyleads to n α=0 S(n, α)b α if and only if b n = n k=0 s(n, k)a k .Preprints (www.