A Note on a Theorem of Schumacher

Antônio Francisco Neto^{1} DEPRO, Escola de Minas Campus Morro do Cruzeiro, UFOP 35400-000 Ouro Preto MG Brazil

                     antfrannet@gmail.com

Abstract

In this paper, we give a short new proof of a recent result due to Schumacher concerning an extension of Faulhaber’s identity for the Bernoulli numbers. Our approach follows from basic manipulations involving the ordinary generating function for the Bernoulli polynomials in the context of the Zeon algebra.

1 Introduction

Zeon algebra [20, Chap. 5], [27, Chap. 2] and Grassmann algebra [5, Chap. 1], [10, Chap. 1], [26, Chap. 3], [27, Chap. 2] are efficient tools towards proving combinatorial identities. In the context of the Zeon algebra, examples include a criterion for ergodicity of Markov chains [13], alternative proofs of Spivey’s identity for Bell numbers [22], the one-variable Faà di Bruno formula [22], identities involving Stirling numbers of the second kind, Bernoulli numbers, and Bernoulli polynomials [23, 24, 25]. Building on ideas from Grassmann algebra we mention, e.g., proofs of theorems of the matrix-tree type [1], representation of the generating function for hyperforests in hypergraphs [4], Cayley-type identities [6], Lindström-Gessel-Viennot lemma, and Schur functions [7].

In this paper, we will give yet another example of the utility of the Zeon algebra by giving a new, simple, and short proof of an extension of Faulhaber’s identity for the Bernoulli numbers [14, Chap. 6] obtained recently by Schumacher [28]. Another compelling feature of our proof is that it does not assume the usual Faulhaber formula a priori, as in the proof given by Schumacher. More precisely, we will show that

1This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) under grant 307211/2015-0.

∑_{i=0}^{⌊x⌋} i^{n} = \frac{x^{n+1}}{n + 1} + (−1)^{n} \frac{B_{n+1}}{n + 1} + \frac{1}{n + 1} ∑_{j=1}^{n+1}(−1)^{j}\binom{n + 1}{j}B_{j}(\{x\})x^{n−j+1},  (1)

using the Zeon algebra [20, 22]. Throughout this work, we let ℕ, ℚ, and ℝ denote the natural, rational, and real numbers, respectively. We define ℕ_{0} := {0}∪ℕ and ℝ_{0}^{+} := {x ∈ ℝ : x ≥ 0}. For x ∈ ℝ_{0}^{+}, we write ⌊x⌋ for the floor of x and {x} for the fractional part of x. In Eq. (1) we take x ∈ ℝ_{0}^{+} and n ∈ ℕ_{0}.

We remark that there are other examples concerning extensions of Faulhaber’s identity in different contexts [2, 3, 8, 9, 11, 12, 15, 16, 17, 18, 19, 21].

Before we continue, we establish the basic underlying algebraic setup needed to give the proof of Eq. (1).

2 Basic definitions: Zeon algebra and the Grassmann-Berezin integral

Definition 1. The Zeon algebra Z_{n} ⊃ ℝ is defined as the associative algebra generated by the collection {ε_{i}}_{i=1}^{n} (n < ∞) and the scalar 1 ∈ ℝ, such that 1ε_{i} = ε_{i} = ε_{i}1, ε_{i}ε_{j} = ε_{j}ε_{i} ∀ i, j and ε_{i}^{2} = 0 ∀ i.

For {i, j, …, k} ⊂ {1, 2, …, n} and ε_{ij···k} ≡ ε_{i}ε_{j} ⋯ ε_{k} the most general element with n generators ε_{i} can be written as (with the convention of sum over repeated indices implicit)

ϕ_{n} = a + a_{i}ε_{i} + a_{ij}ε_{ij} + ⋯ + a_{12···n}ε_{12···n} = ∑_{i∈2^{[n]}} a_{i}ε_{i},

with a, a_{i}, a_{ij}, …, a_{12···n} ∈ ℝ, 2^{[n]} being the power set of [n] := {1, 2, …, n}, and 1 ≤ i < j < ⋯ ≤ n. We refer to a as the body of ϕ_{n} and write b(ϕ_{n}) = a and to ϕ_{n} − a as the soul such that s(ϕ_{n}) = ϕ_{n} − a. This terminology is borrowed from the literature on superanalysis [10, Chap. 1].

Definition 2. The Grassmann-Berezin integral on Z_{n}, denoted by ∫, is the linear functional ∫ : Z_{n} → ℝ such that (we use throughout this work the compact notation dν_{n} := dε_{n} ⋯ dε_{1})

dε_{i}dε_{j} = dε_{j}dε_{i}, ∫ ϕ_{n}(ε̂_{i})dε_{i} = 0, and ∫ ϕ_{n}(ε̂_{i})ε_{i}dε_{i} = ϕ_{n}(ε̂_{i}),

where ϕ_{n}(ε̂_{i}) means any element of Z_{n} with no dependence on ε_{i}. Multiple integrals are iterated integrals, i.e.,

∫ f(ϕ_{n})dν_{n} = ∫ ⋯ (∫ (∫ f(ϕ_{n})dε_{n})dε_{n−1}) ⋯ dε_{1}.

We now extend some of the constructions of previous work on Zeons and Bernoulli numbers [24] to the context of Bernoulli polynomials [14, Chap. 7], [29, Chap. 4]. Let us write ℚ[[x, z]] for the ring of formal power series in the variables x and z over ℚ. We recall the generating function for the Bernoulli polynomials B_{j}(x) in ℚ[[x, z]], i.e.,

\frac{e^{xz}}{∑{∞ \atop i=0} \frac{z^{i}}{(i+1)!}} = \frac{ze^{xz}}{e^{z} − 1} = ∑_{j=0}^{∞} B_{j}(x)\frac{z^{j}}{j!}  (2)

and, making the change z ↦ −z in Eq. (2), we get

\frac{e^{(1−x)z}}{∑{∞ \atop i=0} \frac{z^{i}}{(i+1)!}} = \frac{ze^{(1−x)z}}{e^{z} − 1} = ∑_{j=0}^{∞} B_{j}(x)\frac{(−z)^{j}}{j!} .  (3)

Note that B_{j}(0) ≡ B_{j} are the Bernoulli numbers.

Following the strategy of our previous work [22, 24], we consider Eqs. (2) and (3) in the context of the Zeon algebra with the replacement z → ϕ_{k} ≡ φ_{k} := ε_{1} + ⋯ + ε_{k}. Therefore, we get

\frac{e^{xφ_{k}}}{∑{k \atop i=0} \frac{φ_{k}^{i}}{(i+1)!}} = ∑_{j=0}^{k} B_{j}(x)\frac{φ_{k}^{j}}{j!}  (4)

and

\frac{e^{(1−x)φ_{k}}}{∑{k \atop i=0} \frac{φ_{k}^{i}}{(i+1)!}} = ∑_{j=0}^{k} B_{j}(x)\frac{(−φ_{k})^{j}}{j!} ,  (5)

using that φ_{k}^{k+1} = 0 ∀ k ≥ 1. We observe that

b (∑_{i=0}^{k} \frac{φ_{k}^{i}}{(i + 1)!}) = 1 ≠ 0  (6)

and, hence, ∑_{i=0}^{k} \frac{φ_{k}^{i}}{(i+1)!} is invertible in Z_{k}.

Now, integrating Eq. (4) in the Zeon algebra and using Definition 2, we get the representation of the Bernoulli polynomials

∫ \frac{e^{xφ_{k}}}{∑{k \atop i=0} \frac{φ_{k}^{i}}{(i+1)!}} dν_{k} = ∑_{j=0}^{k} \frac{B_{j}(x)}{j!} ∫ φ_{k}^{j}dν_{k} = B_{k}(x)

∀ k ≥ 1.

Similarly, integrating Eq. (5) in the Zeon algebra, we get

∫ \frac{e^{(1−x)φ_{k}}}{∑{k \atop i=0} \frac{φ_{k}^{i}}{(i+1)!}} dν_{k} = ∑_{j=0}^{k}(−1)^{j} \frac{B_{j}(x)}{j!} ∫ φ_{k}^{j}dν_{k} = (−1)^{k}B_{k}(x)  (7)

∀ k ≥ 1.

3 Proof of Eq. (1)

We are now ready to prove Eq. (1). We start with the following identity

e^{(⌊x⌋+1)φ_{n+1}} − e^{φ_{n+1}} = ∑_{i=1}^{⌊x⌋} e^{iφ_{n+1}} (e^{φ_{n+1}} − 1) = φ_{n+1} ∑_{i=1}^{⌊x⌋} e^{iφ_{n+1}} (∑_{j=0}^{n+1} \frac{φ_{n+1}^{j}}{(j + 1)!}) .

Note that Eq. (6) can be used and we can write

\frac{e^{(⌊x⌋+1)φ_{n+1}} − e^{φ_{n+1}}}{∑{n+1 \atop j=0} \frac{φ_{n+1}^{j}}{(j+1)!}} = φ_{n+1} ∑_{i=1}^{⌊x⌋} e^{iφ_{n+1}}.

Next, using the Grassmann-Berezin integration of Definition 2, we have

∫ \frac{e^{(⌊x⌋+1)φ_{n+1}} − e^{φ_{n+1}}}{∑{n+1 \atop j=0} \frac{φ_{n+1}^{j}}{(j+1)!}} dν_{n+1} = ∫ φ_{n+1} ∑_{i=1}^{⌊x⌋} e^{iφ_{n+1}}dν_{n+1}.  (8)

We observe that

∫ φ_{n+1} ∑_{i=1}^{⌊x⌋} e^{iφ_{n+1}}dν_{n+1} = (n + 1) ∑_{i=1}^{⌊x⌋} ∫ e^{iφ_{n}}dν_{n} = (n + 1) ∑_{i=1}^{⌊x⌋} i^{n}.  (9)

Using Eq. (7) and x = ⌊x⌋ + {x}, we obtain

∫ \frac{e^{(⌊x⌋+1)φ_{n+1}}}{∑{n+1 \atop j=0} \frac{φ_{n+1}^{j}}{(j+1)!}} dν_{n+1} = ∫ \frac{e^{(x−\{x\}+1)φ_{n+1}}}{∑{n+1 \atop j=0} \frac{φ_{n+1}^{j}}{(j+1)!}} dν_{n+1}
= ∑_{k=0}^{n+1} \frac{x^{k}}{k!} ∫ \frac{e^{(1−\{x\})φ_{n+1}}}{∑{n+1 \atop j=0} \frac{φ_{n+1}^{j}}{(j+1)!}} φ_{n+1}^{k}dν_{n+1}
= x^{n+1} + ∑_{k=0}^{n} \binom{n + 1}{k}x^{k} ∫ \frac{e^{(1−\{x\})φ_{n−k+1}}}{∑{n−k+1 \atop j=0} \frac{φ_{n−k+1}^{j}}{(j+1)!}} dν_{n−k+1}
= x^{n+1} + ∑_{k=0}^{n} \binom{n + 1}{k}(−1)^{n−k+1}B_{n−k+1}(\{x\})x^{k}
= x^{n+1} + ∑_{k=1}^{n+1} \binom{n + 1}{k}(−1)^{k}B_{k}(\{x\})x^{n−k+1}  (10)

making the change of variables n − k + 1 ↦ k to obtain the last equality. Finally, taking x = 0 in Eq. (7), we get

∫ \frac{e^{φ_{n+1}}}{∑{n+1 \atop j=0} \frac{φ_{n+1}^{j}}{(j+1)!}} dν_{n+1} = (−1)^{n+1}B_{n+1}.  (11)

Collecting the results in Eqs. (9), (10), (11), and going back to Eq. (8), we arrive at the desired result, i.e., Eq. (1).

4 Acknowledgments

The author thanks the anonymous referee for suggestions that improved the paper.

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2010 Mathematics Subject Classification: Primary 11B68; Secondary 33B10, 05A15, 05A19. Keywords: Zeon algebra, Berezin integration, Bernoulli number, generating function, Faulhaber’s formula.

(Concerned with sequences \underline{A027641} and \underline{A027642}.)

Received June 19 2016; revised version received October 1 2016. Published in Journal of Integer Sequences, October 10 2016.

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