The Dual of Spivey’s Bell Number Identity from Zeon Algebra

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 new short proof of the dual of Spivey’s Bell number identity due to Mező. Our approach follows from basic manipulations involving a fundamental identity representing factorials in the Zeon algebra. This work, along with a previous one due to the author and dos Anjos, shows that Spivey’s and Mező’s identities have at their root a common underlying algebraic origin.

1 Introduction

In this paper, we will give a new, simple, and short proof of the dual of Spivey’s Bell number identity obtained by Mező [5, Chap. 3], [6]. We mention that the original proofs of the identities due to Spivey and Mező are combinatorial in nature [3, 6, 11]. Indeed, the proof given by Mező is constructed by considering the enumeration of the permutations of m + n elements in terms of the number of k-permutations of n and the number of permutations of m with j disjoint cycles. This work, along with a previous work concerning the proof of Spivey’s identity from Zeons [7], adds another point of view in the origin of Spivey’s and Mező’s results which is algebraic in nature, as a direct consequence of the use of the Zeon algebra [5, Chap. 5], [7], [10, Chap. 2]. Throughout this work we let ℝ denote the real numbers and ℕ the positive integers.

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

More precisely, if m, n ∈ ℕ, we will show that

(m + n)! = ∑_{k=0}^{n} ∑_{j=0}^{m} m\overline{n−k}\left[{m \atop j}\right]\binom{n}{k}k!  (1)

using the Zeon algebra. In Eq. (1) a\overline{n} = a(a + 1) ⋯ (a + n − 1) is the Pochhammer symbol with a ∈ ℝ, \binom{n}{k} = n!/(k!(n − k)!) is the binomial coefficient, and \left[{m \atop j}\right] are the unsigned Stirling numbers of the first kind [4, Chap. 6], [12, Eq. (3.5.3)].

For completeness, in the next section, we introduce the tools needed to give the proof of Eq. (1). More precisely, we give the definition of the Zeon algebra and the Grassmann-Berezin integral in the Zeon algebra.

2 Brief review of the Zeon algebra and Grassmann-Berezin integral in the Zeon algebra

Definition 1. The Zeon algebra Z_{n} ⊃ ℝ is defined as the associative and commutative algebra generated by the collection {ε_{i}}_{i=1}^{n} (n < ∞) of nilpotent elements and the scalar 1 ∈ ℝ, which is the identity of the algebra.

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 with the convention of sum over repeated indices implicit and taking ε_{∅} = 1 as

ϑ_{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 ϑ_{n} and write b(ϑ_{n}) := a and to ϑ_{n} − a as the soul such that s(ϑ_{n}) := ϑ_{n} − a. Note that s^{n+1}(ϑ_{n}) = 0.

As described in previous work [7] a real analytic function f can be extended to the realm of the Zeon algebra taking

f(ϑ_{n}) := ∑_{i=0}^{n} \frac{f^{(i)}(b(ϑ_{n}))}{i!}s^{i}(ϑ_{n}) = ∑_{i=0}^{n} \frac{f^{(i)}(a)}{i!}s^{i}(ϑ_{n}),  (2)

where f^{(i)}(a) = d^{i}f(x)/dx^{i}|_{x=a} is the i-th ordinary derivative of f(x) at a. Note that f(a + s(ϑ_{n}))|_{s(ϑ_{n})=0} = f(a) and f(a + s(ϑ_{n})) ∈ Z_{n}, because s^{n+1}(ϑ_{n}) = 0.

We consider specific examples of Eq. (2) important for this work. If a ≠ 0, then ϑ_{n} is invertible and the inverse is given by

ϑ_{n}^{−1} := \frac{1}{a} (1 − \frac{s(ϑ_{n})}{a} + \frac{s^{2}(ϑ_{n})}{a^{2}} + ⋯ + (−1)^{n} \frac{s^{n}(ϑ_{n})}{a^{n}}) .  (3)

Next, we will need

e^{ϑ_{n}} := e^{a} ∑_{m=0}^{n} \frac{s^{m} (ϑ_{n})}{m!}.  (4)

As a final example, for ϑ_{n} = 1 + s(ϑ_{n}), we consider

ln (1 + s (ϑ_{n})) := ∑_{m=1}^{n}(−1)^{m+1} \frac{s^{m}(ϑ_{n})}{m},  (5)

where the right-hand side of Eq. (5) is the solution of the equation

e^{ln(1+s(ϑ_{n}))} ≡ 1 + s(ϑ_{n}).

We give some examples of the functions considered in Eqs. (3), (4), and (5) for n = 3. We have

\frac{1}{1 + \frac{1}{2}ε_{2} + \sqrt{3}ε_{13} − 5ε_{123}} = 1 − \frac{1}{2}ε_{2} − \sqrt{3}ε_{13} + (5 + \sqrt{3}) ε_{123},

e^{1+ \frac{1}{2} ε_{2}+\sqrt{3}ε_{13}−5ε_{123}} = e (1 + \frac{1}{2}ε_{2} + \sqrt{3}ε_{13} + (\frac{\sqrt{3}}{2} − 5) ε_{123}) ,

and

ln (1 + \frac{1}{2}ε_{2} + \sqrt{3}ε_{13} − 5ε_{123}) = \frac{1}{2}ε_{2} + \sqrt{3}ε_{13} − (5 + \frac{\sqrt{3}}{2}) ε_{123}.

Note that e^{\frac{1}{2} ε_{2}+\sqrt{3}ε_{13}−(5+ \frac{\sqrt{3}}{2} )ε_{123}} = 1 + \frac{1}{2}ε_{2} + \sqrt{3}ε_{13} − 5ε_{123}.

Definition 2. The Grassmann-Berezin integral on Z_{n}, denoted by ∫, is the linear functional ∫ : Z_{n} → ℝ such that

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

where ϑ_{n}(ε̂_{i}) means any element of Z_{n} with no dependence on ε_{i}. We use throughout this work the compact notation dν_{n} := dε_{n} ⋯ dε_{1}. Multiple integrals are iterated integrals, i.e.,

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

The standard literature on Grassmann algebra comprises, e.g., the work of Berezin [1, Chap. 1], DeWitt [2, Chap. 1], and Rogers [9, Chap. 3].

Some examples of the integration in Definition 2 are given below. We have

∫ \frac{1}{1 + \frac{1}{2}ε_{2} + \sqrt{3}ε_{13} − 5ε_{123}} dε_{1} = −\sqrt{3}ε_{3} + (5 + \sqrt{3}) ε_{23}

and

∫ e^{1+ \frac{1}{2} ε_{2}+\sqrt{3}ε_{13}−5ε_{123}}dε_{1}dε_{3} = e (\sqrt{3} + (\frac{\sqrt{3}}{2} − 5) ε_{2}) .

A particularly simple result which follows from the multinomial theorem is

∫ φ_{n}^{n}dν_{n} = n!  (6)

with ϑ_{n} = φ_{n} := ε_{1} + ⋯ + ε_{n}.

3 Proof of Eq. (1)

We are now ready to prove Eq. (1). Let us take ϕ_{m} = ϵ_{1} + ⋯ + ϵ_{m} with ϵ_{i} := ε_{n+i}, i = 1, …, m, such that φ_{n+m} = ϕ_{m} + φ_{n} ∈ Z_{m+n}. We start with the following identity

(m + n)! = ∫ (ϕ_{m} + φ_{n})^{m+n}dμ_{m}dν_{n} = ∫ ∑_{k=0}^{m+n}(ϕ_{m} + φ_{n})^{k}dμ_{m}dν_{n} = ∫ \frac{1}{1 − ϕ_{m} − φ_{n}} dμ_{m}dν_{n},

(7)

using Eqs. (3) and (6). Next, we observe that

(m + n)! = ∫ \frac{1}{1 − φ_{n}} \frac{1}{1 − \frac{ϕ_{m}}{1−φ_{n}}} dμ_{m}dν_{n}
= ∑_{k=0}^{n} ∫ φ_{n}^{k} \frac{1}{1 − \frac{ϕ_{m}}{1−φ_{n}}} dμ_{m}dν_{n}
= ∫ \frac{1}{1 − \frac{ϕ_{m}}{1−φ_{n}}} dμ_{m}dν_{n} + ∑_{k=1}^{n} ∑_{1≤j_{1}<···<j_{k}≤n} k! ∫ ε_{j_{1}···j_{k}} \frac{1}{1 − \frac{ϕ_{m}}{1−φ_{n}}} dμ_{m}dν_{n}

= ∑_{k=0}^{n} \binom{n}{k}k! ∫ \frac{1}{1 − \frac{ϕ_{m}}{1−φ_{n−k}}} dμ_{m}dν_{n−k}
= ∑_{k=0}^{n} \binom{n}{k}k! ∫ \frac{1 − φ_{n−k}}{1 − ϕ_{m} − φ_{n−k}} dμ_{m}dν_{n−k}
= ∑_{k=0}^{n} \binom{n}{k}k! (∫ \frac{1}{1 − ϕ_{m} − φ_{n−k}} dμ_{m}dν_{n−k} − ∫ \frac{n − k}{1 − ϕ_{m} − φ_{n−k−1}} dμ_{m}dν_{n−k−1})
= ∑_{k=0}^{n} \binom{n}{k}k! ((m + n − k)! − (n − k) (m + n − k − 1)!)
= ∑_{k=0}^{n} \binom{n}{k}k!m (m + n − k − 1)!,

using Eq. (7). Finally, recalling the definition of m\overline{n−k} in Eq. (1), we obtain

(m + n)! = m! ∑_{k=0}^{n} \binom{n}{k}k!m\overline{n−k}.  (8)

Now, we consider the unsigned Stirling numbers of the first kind \left[{n \atop j}\right] given by the generating function [12, Eq. (3.5.3)]

\frac{1}{j!} (ln \frac{1}{1 − x})^{j} = ∑_{n=j}^{∞} \left[{n \atop j}\right]\frac{x^{n}}{n!} .  (9)

Using Eqs. (5) and (6) we arrive at

\left[{m \atop j}\right] = \frac{1}{j!} ∫ (ln \frac{1}{1 − ϕ_{m}} )^{j} dμ_{m},

extending Eq. (9) to the context of Zeons by following previous work [7, 8]. Note that

∑_{j=0}^{m} \left[{m \atop j}\right] = ∫ e^{ln}^{(^{\frac{1}{1−ϕm}})}dμ_{m} = ∫ \frac{1}{1 − ϕ_{m}} dμ_{m} = m!,  (10)

using Eqs. (3), (4), (5), and (6).

Finally, from Eqs. (8) and (10), we arrive at Eq. (1).

References

[1] F. A. Berezin, Introduction to Superanalysis, Reidel Publishing Company, 1987.

[2] B. DeWitt, Supermanifolds, Cambridge University Press, 1992.

[3] H. W. Gould and J. Quaintance, Implications of Spivey’s Bell number formula, J. Integer Seq. 11 (2008), Article 08.3.7.

[4] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, second edition, 1994.

[5] T. Mansour and M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, Chapman and Hall/CRC Press, 2015.

[6] I. Mező, The dual of Spivey’s Bell number formula, J. Integer Seq. 15 (2012), Article 12.2.4.

[7] A. F. Neto and P. H. R. dos Anjos, Zeon algebra and combinatorial identities, SIAM Rev. 56 (2014), 353–370.

[8] A. F. Neto, A note on a theorem of Guo, Mező, and Qi, J. Integer Seq. 19 (2016), Article 16.4.8.

[9] A. Rogers, Supermanifolds: Theory and Applications, World Scientific Publishing, 2007.

[10] R. Schott and G. S. Staples, Operator Calculus on Graphs, Imperial College Press, 2012.

[11] M. Z. Spivey, A generalized recurrence for Bell numbers, J. Integer Seq. 11 (2008), Article 08.2.5.

[12] H. S. Wilf, Generatingfunctionology, Academic Press, New York, 1990. Available at

    http://www.math.upenn.edu/~wilf/DownldGF.html.

2010 Mathematics Subject Classification: Primary 11B73; Secondary 33B10, 05A15, 05A19. Keywords: Zeon algebra, Berezin integration, Stirling number of the first kind, Pochhammer symbol, binomial coefficient, generating function.

(Concerned with sequences \underline{A000142} and \underline{A132393}.)

Received August 1 2016; revised version received November 26 2016. Published in Journal of Integer Sequences, December 27 2016.

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