Spivey’s Bell Number Formula Revisited

Mahid M. Mangontarum Department of Mathematics Mindanao State University—Main Campus Marawi City 9700 Philippines

                    mmangontarum@yahoo.com
               mangontarum.mahid@msumain.edu.ph

Abstract This paper introduces an alternative form of the derivation of Spivey’s Bell number formula, which involves the q-Boson operators a and a^{†}. Furthermore, a similar formula for the case of the (q, r)-Dowling polynomials is obtained, and is shown to produce a generalization of the latter.

1 Introduction

Consider the Stirling numbers of the second kind, denoted by \left\{{m \atop j}\right\}, which appear as coefficients in the expansion of

t^{n} = ∑_{k=0}^{n} \left\{{n \atop k}\right\}(t)_{k},  (1)

where (t)_{k} = t(t − 1)(t − 2) ⋯ (t − k + 1).The Bell numbers, denoted by B_{n}, are defined by

B_{n} = ∑_{j=0}^{n} \left\{{n \atop j}\right\}  (2)

and are known to satisfy the recurrence relation

B_{n+1} = ∑_{k=0}^{n} \binom{n}{k}B_{k}.  (3)

In 2008, Spivey [13] obtained a remarkable formula which unifies the defining relation in (2) and the identity (3). The said formula is given by

B_{n+m} = ∑_{k=0}^{n} ∑_{j=0}^{m} j^{n−k}\left\{{m \atop j}\right\}\binom{n}{k}B_{k}  (4)

and is popularly known as “Spivey’s Bell number formula”. Equation (4) was proved in [13] using a combinatorial approach involving partition of sets. Different proofs and extensions of (4) were later on studied by several authors. For instance, a proof which made use of generating functions was done by Gould and Quaintance [5] which was then generalized by Xu [14] using Hsu and Shuie’s [6] generalized Stirling numbers. Belbachir and Mihoubi [2] presented a proof that involves decomposition of the Bell polynomials into a certain polynomial basis. Mező [12] obtained a generalization of the Spivey’s formula in terms of the r-Bell polynomials via combinatorial approach. The notion of dual of (4) was also presented in the same paper. On the other hand, the work of Katriel [7] involved the use of the operator X satisfying

DX − qXD = 1,  (5)

where D is the q-derivative defined by

Df(x) = \frac{f(qx) − f(x)}{x(q − 1)}.  (6)

For the sake of clarity and brevity, this method will be referred to as “Katriel’s proof”.

Now, aside from being implicitly implied in Katriel’s proof, none of the previouslymentioned studies considered establishing q-analogues. It is, henceforth, the main purpose of this paper to obtain a generalized q-analogue of Spivey’s Bell number formula.

2 Alternative form of “Katriel’s proof”

We direct our attention to the q-Boson operators a and a^{†} satisfying the commutation relation

[a, a^{†}]_{q} = aa^{†} − qa^{†}a = 1  (7)

(see [1]). We define the Fock space (or Fock states) by the basis {|s⟩; s = 0, 1, 2, …} so that the relations a |s⟩ = \sqrt{[s]_{q}} |s − 1⟩ and a^{†} |s⟩ = \sqrt{[s + 1]_{q}} |s + 1⟩ form a representation that satisfies (7). The operators a^{†}a and (a^{†})^{k}a^{k}, when acting on |s⟩, yield

a^{†}a |s⟩ = [s]_{q} |s⟩  (8)

and

(a^{†})^{k}a^{k} |s⟩ = [s]_{q,k} |s⟩ ,  (9)

respectively, where [s]_{q} = \frac{q^{s}−1}{q−1} and [s]_{q,k} = [s]_{q}[s − 1]_{q}[s − 2]_{q} ⋯ [s − k + 1]_{q}. Hence, the q-Stirling numbers of the second kind \left\{{n \atop k}\right\}_{q} [3] can be defined alternatively as

(a^{†}a)^{n} = ∑_{k=1}^{n} \left\{{n \atop k}\right\}_{q}(a^{†})^{k}a^{k}.  (10)

From (7), it is clear that

[a, (a^{†})^{k}]_{q^{k}} = [a, (a^{†})^{k−1}]_{q^{k−1}}a^{†} + q^{k−1}(a^{†})^{k−1}[a, a^{†}]_{q},  (11)

and by induction on k, we have

[a, (a^{†})^{k}]_{q^{k}} = [k]_{q}(a^{†})^{k−1}.  (12)

Since a |0⟩ = 0, then by (12),

a(a^{†})^{ℓ} |0⟩ = [a, (a^{†})^{ℓ}]_{q^{ℓ}} |0⟩
= [ℓ]_{q}(a^{†})^{ℓ−1} |0⟩ .

Moreover,

a^{k}(a^{†})^{ℓ} |0⟩ = \frac{[ℓ]_{q}!}{[ℓ − k]_{q}!}(a^{†})^{ℓ−k} |0⟩ ,  (13)

for k ≤ ℓ and

a^{k}(a^{†})^{ℓ} |0⟩ = 0,  (14)

for k > ℓ. Finally,

a^{k}e_{q}(xa^{†}) |0⟩ = x^{k}e_{q}(xa^{†}) |0⟩ ,  (15)

where e_{q}(xa^{†}) is the q-exponential function defined by

e_{q}(t) = ∑_{ℓ=0}^{∞} \frac{t^{ℓ}}{[ℓ]_{q}!}.  (16)

Applying (15) to (10) yields

(a^{†}a)^{n}e_{q}(ta^{†}) |0⟩ = B_{n,q}(ta^{†})e_{q}(ta^{†}) |0⟩ ,  (17)

where B_{n,q}(ta^{†}) denotes the q-Bell polynomials defined by

B_{n,q}(t) = ∑_{k=0}^{n} \left\{{n \atop k}\right\}_{q}t^{k}.  (18)

Let x = ta^{†} so that

(a^{†}a)^{n}e_{q}(x) |0⟩ = B_{n,q}(x)e_{q}(x) |0⟩ .  (19)

Before proceeding, note that by definition,

[a, (a^{†})^{k}]_{q^{k}} = a(a^{†})^{k} − q^{k}(a^{†})^{k}a.  (20)

By (12),

a(a^{†})^{k} − q^{k}(a^{†})^{k}a = [k]_{q}(a^{†})^{k−1}
a(a^{†})^{k} = q^{k}(a^{†})^{k}a + [k]_{q}(a^{†})^{k−1}.

This can be further expressed as

(a^{†}a)(a^{†})^{k} = (a^{†})^{k}([k]_{q} + q^{k}(a^{†}a)).  (21)

Now, we have

(a^{†}a)^{n+m} = (a^{†}a)^{n} ∑_{j=0}^{m} \left\{{m \atop j}\right\}_{q}(a^{†})^{j}a^{j}

= ∑_{j=0}^{m} \left\{{m \atop j}\right\}_{q}(a^{†})^{j}([j]_{q} + q^{j}(a^{†}a))^{n}a^{j}

= ∑_{j=0}^{m} ∑_{k=0}^{n} \left\{{m \atop j}\right\}_{q}\binom{n}{k}[j]_{q}^{n−k}q^{jk}(a^{†})^{j}(a^{†}a)^{k}a^{j}.

Multiplying both sides with e_{q}(x) |0⟩ makes the left-hand side

(a^{†}a)^{n+m}e_{q}(x) |0⟩ = B_{n+m,q}(x)e_{q}(x) |0⟩ ,  (22)

while the right-hand side becomes

∑_{j=0}^{m} ∑_{k=0}^{n} \left\{{m \atop j}\right\}_{q}\binom{n}{k}[j]_{q}^{n−k}q^{jk}(a^{†})^{j}(a^{†}a)^{k}e_{q}(x) |0⟩ a^{j} = ∑_{j=0}^{m} ∑_{k=0}^{n} \left\{{m \atop j}\right\}_{q}\binom{n}{k}[j]_{q}^{n−k}q^{jk}
B_{k,q}(x)e_{q}(x) |0⟩ (a^{†})^{j}a^{j}.

Dividing both sides by e_{q}(x) |0⟩ and using (9) gives

B_{n+m,q}(x) = ∑_{j=0}^{m} ∑_{k=0}^{n} \left\{{m \atop j}\right\}_{q}\binom{n}{k}[j]_{q}^{n−k}q^{jk}B_{k,q}(x)[x]_{q,j}.  (23)

As q → 1, we obtain a polynomial version of Spivey’s Bell number formula which, in return, reduces to (4) when we set x = 1.

It is important to emphasize that this is not a new proof, but an alternative form of Katriel’s proof, since the operators a, a^{†} and the operators X, D generate isomorphic algebras.

3 A generalization of Spivey’s Bell number formula

The main result of this paper is the following identity:

D_{m,r,q}(n + ℓ, x) = ∑_{j=0}^{ℓ} ∑_{k=0}^{n} m^{j}W_{m,r,q}(ℓ, j)\binom{n}{k}(m[j]_{q} + r)^{n−k}q^{jk}D_{m,0,q}(k, x)[x]_{q,j}. (24)

Here, D_{m,r,q}(n, x) is a (q, r)-Dowling polynomial defined previously by the author and Katriel [9] as

D_{m,r,q}(n, x) = ∑_{k=0}^{n} W_{m,r,q}(n, k)x^{k},  (25)

where W_{m,r,q}(n, k) is the (q, r)-Whitney numbers of the second kind. Several properties of D_{m,r,q}(n, x) can be seen in [8, 9].

To derive (24), we first multiply both sides of (21) by m and then add r(a^{†})^{k} to yield

(ma^{†}a + r)(a^{†})^{k} = (a^{†})^{k}(m[k]_{q} + r + mq^{k}a^{†}a).  (26)

Also, multiplying both sides of the defining relation in [9, Equation 16] by e_{q}(ta^{†}) |0⟩ and applying (15) yields

(ma^{†}a + r)^{n}e_{q}(ta^{†}) |0⟩ = ∑_{k=0}^{n} m^{k}W_{m,r,q}(n, k)(a^{†})^{k}a^{k}e_{q}(ta^{†}) |0⟩

= ∑_{k=0}^{n} m^{k}W_{m,r,q}(n, k)(a^{†})^{k}t^{k}e_{q}(ta^{†}) |0⟩
= D_{m,r,q}(n, mta^{†})e_{q}(ta^{†}) |0⟩ .

Now, by (26),

(ma^{†}a + r)^{n+ℓ} = ∑_{j=0}^{ℓ} m^{j}W_{m,r,q}(ℓ, j)(ma^{†}a + r)^{n}(a^{†})^{j}a^{j}

= ∑_{j=0}^{ℓ} m^{j}W_{m,r,q}(ℓ, j)(a^{†})^{j}(m[j]_{q} + r + mq^{j}a^{†}a)^{n}a^{j}

= ∑_{j=0}^{ℓ} ∑_{k=0}^{n} m^{j+k}W_{m,r,q}(ℓ, j)\binom{n}{k}(a^{†})^{j}(m[j]_{q} + r)^{n−k}q^{kj}(a^{†}a)^{k}a^{j}.

Applying this expression to the operator identity e_{q}(ta^{†}) |0⟩, combining with the previous equation, using (9), (19) and W_{m,0,q}(k, i) = m^{k−i}\left\{{k \atop i}\right\}_{q} (see [9, Equation 18]), and then dividing both sides of the resulting identity by e_{q}(ta^{†}) |0⟩ completes the derivation.

4 Remarks

Since W_{1,0,q}(ℓ, j) = \left\{{ℓ \atop j}\right\}_{q}, then by setting x = 1, m = 1 and r = 0, we have

D_{1,0,q}(n + ℓ, 1) = ∑_{j=0}^{ℓ} ∑_{k=0}^{n} \left\{{ℓ \atop j}\right\}_{q}\binom{n}{k}[j]_{q}^{n−k}q^{jk}B_{k,q},  (27)

where B_{k,q} := B_{k,q}(1). This is a q-analogue of (4) which was first obtained by Katriel [7]. On the other hand, setting x = 1 and then taking the limit of (24) as q → 1 provides a generalization of Spivey’s Bell number formula in terms of the r-Whitney numbers of the second kind, denoted by W_{m,r}(ℓ, j), and the r-Dowling numbers, denoted by D_{m,r}(n), (see [4, 11]), given by

D_{m,r}(n + ℓ) = ∑_{j=0}^{ℓ} ∑_{k=0}^{n} m^{j}W_{m,r}(ℓ, j)\binom{n}{k}(mj + r)^{n−k}D_{m,0}(k).  (28)

In a recent paper, Mansour et al. [10] obtained the following generalization of Spivey’s Bell number formula:

D_{p,q}(a + b; x) = ∑_{i=0}^{a} ∑_{j=0}^{b} ∑_{ℓ=0}^{j}(mq^{i})^{j−ℓ}x^{i+ℓ}\binom{b}{j} ([r]_{p} + m[i]_{q})^{b−j} W_{p,q}(a, i)S_{q}(j, ℓ).  (29)

Here, D_{p,q}(n; x) and W_{p,q}(n, k) denote the (p, q)-analogues of the r-Dowling polynomials and the r-Whitney numbers of the second kind, respectively. The (p, q)-analogues are natural generalizations of q-analogues. However, since the manner by which the numbers W_{m,r,q}(n, k) were defined in [9] differs from the work of Mansour et al. [10], the main result of this paper is not generalized by (29).

5 Acknowledgment

The author is very thankful to the editor-in-chief and to the referee(s) for carefully reading the paper. Their comments and suggestions were very helpful. Special thanks also to Dr. Jacob Katriel for his insights on q-Boson operators. This paper is dedicated to my family and all other victims of the war in Marawi.

References

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

[14] A. Xu, Extensions of Spivey’s Bell number formula, Electron. J. Combin. 19 (2012), #P6.

2010 Mathematics Subject Classification: Primary 11B83; Secondary 11B73, 05A30. Keywords: Spivey’s Bell number formula, (q, r)-Whitney number, q-Boson operator.

(Concerned with sequences \underline{A000110}, \underline{A003575}, \underline{A008275}, and \underline{A008277}.)

Received September 23 2017; revised version received November 27 2017. Published in Journal of Integer Sequences, December 20 2017.

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