On Engel’s Inequality for Bell Numbers

Horst Alzer Morsbacher Straße 10 51545 Waldbröl Germany

                        h.alzer@gmx.de

Abstract We prove that for all integers n ≥ 2 the expression B_{n−1}B_{n+1} − B_{n}^{2} can be represented as an infinite series with nonnegative terms. Here B_{k} denotes the k-th Bell number. It follows that the sequence (B_{n})_{n≥0} is strictly log-convex. This result refines Engel’s inequality B_{n}^{2} ≤ B_{n−1}B_{n+1}.

1 Introduction

A partition of a set S with n elements is a collection of nonempty, pairwise disjoint subsets whose union is equal to S. The Bell number B_{n}, named after the British mathematician Eric T. Bell, is the number of partitions of S. The first few numbers are

B_{0} = 1, B_{1} = 1, B_{2} = 2, B_{3} = 5, B_{4} = 15, B_{5} = 52.

The Bell numbers are given by the exponential generating function

exp(e^{x} − 1) = ∑_{n=0}^{∞} \frac{B_{n}}{n!} x^{n}

and they satisfy the recurrence relation

B_{n+1} = ∑_{k=0}^{n} \binom{n}{k}B_{k} (n ≥ 0).

Moreover, the Bell numbers can be expressed in terms of the Stirling numbers of the second kind,

B_{n} = ∑_{k=1}^{n} S(n, k) (n ≥ 1).

The following remarkable series representation is due to Dobiński [6],

B_{n} = \frac{1}{e} ∑_{k=0}^{∞} \frac{k^{n}}{k!} (n ≥ 0).  (1)

In 1994, Engel [7] proved that the sequence (B_{n})_{n≥0} is log-convex, that is,

B_{n}^{2} ≤ B_{n−1}B_{n+1} (n ≥ 1).  (2)

Canfield [5], Asai, Kubo, and Kuo [1] and Bouroubi [3] published further proofs of (2). For more information and references on Bell numbers we refer to [4] and [8].

Richard E. Bellman (1920–1984), who was one of the leading mathematicians in the field of inequalities, pointed out that “every inequality should come from an equality which makes the inequality obvious” [2, p. 449]. In view of this statement it is natural to ask whether Engel’s inequality is a consequence of an equality. It is the aim of this note to give an affirmative answer to this question. In particular, we prove that for all n ≥ 1 strict inequality holds in (2). We show that an application of Dobiński’s formula leads to the following result.

2 The main result

Theorem 1. For all natural numbers n ≥ 2 we have

B_{n−1}B_{n+1} − B_{n}^{2} = \frac{1}{2e^{2}} ∑_{k=2}^{∞} ∑_{j=1}^{k−1} \frac{j^{n−1}(k − j)^{n−1}}{j!(k − j)!} (k − 2j)^{2}.  (3)

Proof. Applying (1) and the Cauchy product for infinite series we obtain for n ≥ 2,

e^{2}(B_{n−1}B_{n+1} − B_{n}^{2}) = ∑_{k=0}^{∞} ∑_{j=0}^{k} \frac{j^{n−1}(k − j)^{n+1}}{j!(k − j)!} − ∑_{k=0}^{∞} ∑_{j=0}^{k} \frac{j^{n}(k − j)^{n}}{j!(k − j)!}

= ∑_{k=2}^{∞}(L_{n}(k) − R_{n}(k))  (4)

with

L_{n}(k) = ∑_{j=1}^{k−1} \frac{j^{n−1}(k − j)^{n+1}}{j!(k − j)!} and R_{n}(k) = ∑_{j=1}^{k−1} \frac{j^{n}(k − j)^{n}}{j!(k − j)!} .

Since

∑_{j=1}^{k−1} \frac{j^{n−1}(k − j)^{n+1}}{j!(k − j)!} = ∑_{j=1}^{k−1} \frac{j^{n+1}(k − j)^{n−1}}{j!(k − j)!} ,

we get

L_{n}(k) − R_{n}(k) = \frac{1}{2} ∑_{j=1}^{k−1} \frac{1}{j!(k − j)!}(j^{n−1}(k − j)^{n+1} + j^{n+1}(k − j)^{n−1} − 2j^{n}(k − j)^{n})

= \frac{1}{2} ∑_{j=1}^{k−1} \frac{j^{n−1}(k − j)^{n−1}}{j!(k − j)!} (k − 2j)^{2}.  (5)

From (4) and (5) we conclude that (3) holds.

Remark 2. Using B_{0}B_{2} − B_{1}^{2} = 1 and (3) reveals that the sequence (B_{n})_{n≥0} is not only log-convex, but even strictly log-convex,

B_{n}^{2} < B_{n−1}B_{n+1} (n ≥ 1).  (6)

Remark 3. Asai, Kubo, and Kuo [1] used Engel’s inequality to prove that

B_{m}B_{n} ≤ B_{m+n} (m, n ≥ 0).

An application of (6) gives for m, n ≥ 1,

B_{m} = ∏_{ν=1}^{m} \frac{B_{ν}}{B_{ν−1}} < ∏_{ν=1}^{m} \frac{B_{n+ν}}{B_{n+ν−1}} = \frac{B_{m+n}}{B_{n}} .

Thus,

B_{m}B_{n} < B_{m+n} (m, n ≥ 1).

References

[1] N. Asai, I. Kubo, and H.-H. Kuo, Bell numbers, log-concavity, and log-convexity, Acta Appl. Math. 63 (2000), 79–87.

[2] R. Bellman, Why study inequalities?, in E. F. Beckenbach, ed., General Inequalities 2, Int. Ser. Numer. Math. 47, Birkhäuser, 1980, p. 449.

[3] S. Bouroubi, Bell numbers and Engel’s conjecture, Rostock. Math. Kolloq. 62 (2007), 61–70.

[4] D. Branson, Stirling numbers and Bell numbers: their role in combinatorics and probability, Math. Scientist 25 (2000), 1–31.

[5] E. R. Canfield, Engel’s inequality for Bell numbers, J. Combin. Th. Ser. A 72 (1995), 184–187.

[6] G. Dobiński, Summierung der Reihe ∑ n^{m}/n! für m = 1, 2, 3, 4, 5, …, Archiv Math. Phys. 61 (1877), 333–336.

[7] K. Engel, On the average rank of an element in a filter of the partition lattice, J. Combin. Th. Ser. A 65 (1994), 67–78.

[8] N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, OEIS Foundation, Inc., 2019. Available at https://oeis.org.

2010 Mathematics Subject Classification: Primary 05A19; Secondary 05A20, 11B73, 26A51.

K eywords: Bell number, Engel’s inequality, strictly log-convex, identity.

(Concerned with sequence \underline{A000110}.)

Received June 26 2019; revised version received August 25 2019. Published in Journal of Integer Sequences, September 25 2019.

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