On q-Boson Operators and q-Analogues of the r-Whitney and r-Dowling Numbers

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

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

Jacob Katriel Department of Chemistry Technion – Israel Institute of Technology Haifa 32000 Israel

                jkatriel@technion.ac.il

Abstract

We define the (q, r)-Whitney numbers of the first and second kinds in terms of the q-Boson operators, and obtain several fundamental properties such as recurrence formulas, orthogonality and inverse relations, and other interesting identities. As a special case, we obtain a q-analogue of the r-Stirling numbers of the first and second kinds. Finally, we define the (q, r)-Dowling polynomials in terms of sums of (q, r)- Whitney numbers of the second kind, and obtain some of their properties.

1 Introduction

The investigation of q-analogues of combinatorial identities has proven to be a rich source of insight as well as of useful generalizations. Some examples of q-analogues are the q-real number, the q-factorial and the q-falling factorial of order r, respectively, given by

[x]_{q} = \frac{q^{x} − 1}{q − 1}, [n]_{q}! = ∏_{i=1}^{n}[i]_{q}, [x]_{q,n} = ∏_{i=0}^{r−1}[x − i]_{q},

for any real number x and non-negative integers n and r, and the q-binomial coefficients (also known as Gaussian polynomials)

\binom{n}{r}_{q} = \frac{[n]_{q}!}{[r]_{q}![n − r]_{q}!} = \frac{[n]_{q,r}}{[r]_{q}!} .

The formulation of q-analogues is not unique, but some choices appear to allow more productive generalizations than others. In the present paper we apply the properties of the q-boson operators as a framework for the generation of q-deformations of a family of combinatorial identities involving the Whitney numbers.

A lattice L in which every element is the join of elements x and y (in L) such that x and y cover the zero element 0, and is semimodular, is called a geometric lattice. Originally, if L is a finite lattice of rank n, then the Whitney numbers w(n, k) and W (n, k) of the first and second kinds of L are defined as the coefficients of the characteristic polynomial and as the number of elements of L of corank k, respectively. Now, Dowling [20] defined a class of these geometric lattices, called Dowling lattice, which is a generalization of the partition lattice. Let Q_{n}(G) be the Dowling lattice of rank n associated to a finite group G of order m > 0. Benoumhani [3] defined the Whitney numbers of the first and second kind of Q_{n}(G), denoted by w_{m}(n, k) and W_{m}(n, k), respectively, in terms of the relations

m^{n}(x)_{n} = ∑_{k=0}^{n} w_{m}(n, k)(mx + 1)^{k}  (1)

and

(mx + 1)^{n} = ∑_{k=0}^{n} m^{k}W_{m}(n, k)(x)_{k},  (2)

where (x)_{n} = x(x − 1) ⋯ (x − n + 1) is the falling factorial of x of order n. Notice that if the group G is the trivial group (m = 1), multiplication of both equations (1) and (2) by (x + 1) yields the horizontal generating functions for the well-known Stirling numbers of the first and second kind [29], denoted by \left[{n \atop k}\right] and \left\{{n \atop k}\right\}, respectively. Hence,

w_{1}(n, k) = \left[{n + 1 \atop k + 1}\right], W_{1}(n, k) = \left\{{n + 1 \atop k + 1}\right\}.

We note that Benoumhani [3, 4] already established the fundamental properties of the numbers w_{m}(n, k) and W_{m}(n, k) while Dowling [20] gave a detailed discussion of geometric lattices. Other generalizations of the Stirling numbers \left[{n \atop k}\right] and \left\{{n \atop k}\right\} were already considered by several authors. For instance, Broder [5] defined the r-Stirling numbers \begin{matrix} \\ [ & {n+r \atop k+r} & ] \end{matrix}_{r} and \begin{matrix} \\ \{ & {n+r \atop k+r} & \} \end{matrix}_{r} of the first and second kind whose relation to the Whitney numbers is stated in equations (21) and (22) below. Belbachir and Bousbaa [2] recently introduced the translated Whitney numbers w̃_{(α)}(n, k) and W̃_{(α)}(n, k) of the first and second kind, which are related to the Stirling numbers via

w̃_{(α)}(n, k) = α^{n−k}\left[{n \atop k}\right], W̃_{(α)}(n, k) = α^{n−k}\left\{{n \atop k}\right\}.

Furthermore, Mező [27] defined the r-Whitney numbers w_{m,r}(n, k) and W_{m,r}(n, k) of the first and second kind as the coefficients in the expressions

m^{n}(x)_{n} = ∑_{k=0}^{n} w_{m,r}(n, k)(mx + r)^{k}  (3)

and

(mx + r)^{n} = ∑_{k=0}^{n} m^{k}W_{m,r}(n, k)(x)_{k}.  (4)

respectively. Further developement of the numbers w_{m,r}(n, k) and W_{m,r}(n, k) were due to Cheon and Jung [7], Merca [26], Corcino et al. [10], Corcino et al. [19], C. B. Corcino and R. B. Corcino [9], and R. B. Corcino and C. B Corcino [14, 15].

Corcino and Hererra [17] introduced the limit of the differences of the generalized factorial

F_{α,γ}(n, k) = lim_{β→0} \frac{[Δ_{t}^{k} (βt + γ|α)_{n}]_{t=0}}{k!β^{k}} ,  (5)

where

(βt + γ|α)_{n} = ∏_{j=0}^{n−1} (βt + γ − jα) , (βt + γ|α)_{0} = 1,  (6)

which is a generalization of the Stirling numbers of the first kind. The numbers F_{α,−γ}(n, k) are actually the r-Whitney numbers of the first kind in (3). That is,

F_{α,−γ}(n, k) = w_{α,γ}(n, k).

Similarly, Corcino [11] defined the (r, β)-Stirling numbers \left⟨{n \atop k}\right⟩_{r,β} as coefficients in

t^{n} = ∑_{k=0}^{n} \binom{\frac{t−r}{β}}{k}β^{k}k!\left⟨{n \atop k}\right⟩_{r,β}.  (7)

The numbers \left⟨{n \atop k}\right⟩_{r,β} are found to be equivalent to the r-Whitney numbers of the second kind in (4). To be precise,

\left⟨{n \atop k}\right⟩_{r,β} = W_{β,r}(n, k).

Corcino et al. [16], and Corcino and Aldema [12] further studied the numbers \left⟨{n \atop k}\right⟩_{r,β}. Recall that the classical Boson operators a and a^{†} satisfy the commutation relation

[a, a^{†}] ≡ aa^{†} − a^{†}a = 1.  (8)

If we define the Fock space by the basis {|s⟩ ; s = 0, 1, 2, …}, to be referred to as Fock states, the relations a|s⟩ = \sqrt{s}|s − 1⟩ and a^{†}|s⟩ = \sqrt{s + 1}|s + 1⟩ form a representation that satisfies the commutation relation (8). The operator n̂ ≡ a^{†}a, when acting on |s⟩, yields

a^{†}a|s⟩ = s|s⟩,

and the operator (a^{†})^{k}a^{k}, when acting on the same state, yields

(a^{†})^{k}a^{k}|s⟩ = (s)_{k}|s⟩.

Let {⟨s| ≡ (|s⟩)^{†} ; s = 0, 1, 2, …} denote the Fock basis of the dual space. Requiring the normalization of the scalar product ⟨0|0⟩ = 1 we note that

⟨s + 1|s + 1⟩ = \frac{1}{s + 1}⟨s|aa^{†}|s⟩ = \frac{1}{s + 1}(⟨s|a^{†}a|s⟩ + ⟨s|s⟩) = ⟨s|s⟩.

Hence, from the normalization of |0 > it follows that all the Fock states are normalized. Moreover, since ⟨s + 1|a^{†}|s⟩ = \sqrt{s + 1}⟨s + 1|s + 1⟩ and (a|s + 1⟩)^{†}|s⟩ = \sqrt{s + 1}⟨s|s⟩, it follows that a^{†} is the Hermitian conjugate of a. That is, a^{†}a is Hermitian. Orthogonality follows from the fact that the Fock states are eigenstates of a^{†}a with distinct eigenvalues.

Hence, the horizontal generating functions of the Stirling numbers \left[{n \atop k}\right] and \left\{{n \atop k}\right\},

(x)_{n} = ∑_{k=0}^{n}(−1)^{n−k}\left[{n \atop k}\right]x^{k}  (9)

and

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

can be expressed as

(a^{†})^{n}a^{n} = ∑_{k=0}^{n}(−1)^{n−k}\left[{n \atop k}\right](a^{†}a)^{k}

and

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

respectively [22].

Now, the defining relations for the r-Whitney numbers, (3) and (4), can be expressed as

m^{n}(a^{†})^{n}a^{n} = ∑_{k=0}^{n} w_{m,r}(n, k)(ma^{†}a + r)^{k}  (10)

and

(ma^{†}a + r)^{n} = ∑_{k=0}^{n} m^{k}W_{m,r}(n, k)(a^{†})^{k}a^{k}.  (11)

Making use of the q-Boson operators [1] that satisfy

[a, a^{†}]_{q} ≡ aa^{†} − qa^{†}a = 1,  (12)

we have

a|s⟩ = \sqrt{[s]_{q}}|s − 1⟩, a^{†}|s⟩ = \sqrt{[s + 1]_{q}}|s + 1⟩,

hence,

a^{†}a|s⟩ = [s]_{q}|s⟩,

and

(a^{†})^{k}a^{k}|s⟩ = [s]_{q,k}|s⟩.

Remark 1. Although we use the same notation for the boson and for the q-boson operators, no confusion should arise because the meaning of these symbols should be clear from the context.

In line with this, the defining relations for Carlitz’s [6] q-Stirling numbers of the first and second kind, \left[{n \atop k}\right]_{q} and \left\{{n \atop k}\right\}_{q}, can be written in the form [22]

(a^{†})^{n}a^{n} = ∑_{k=1}^{n}(−1)^{n−k}\left[{n \atop k}\right]_{q}(a^{†}a)^{k}  (13)

and

(a^{†}a)^{n} = ∑_{k=1}^{n} \left\{{n \atop k}\right\}_{q}(a^{†})^{k}a^{k},  (14)

respectively.

We define q-analogues for the Whitney numbers w_{m,r}(n, k) and W_{m,r}(n, k) via the same pattern as in (13) and (14).

2 (q, r)-Whitney numbers

Definition 2. For non-negative integers n and k and complex numbers r and m, the (q, r)- Whitney numbers of the first and second kind, denoted by w_{m,r,q}(n, k) and W_{m,r,q}(n, k), respectively, are defined by

m^{n}(a^{†})^{n}a^{n} = ∑_{k=0}^{n} w_{m,r,q}(n, k)(ma^{†}a + r)^{k}  (15)

and

(ma^{†}a + r)^{n} = ∑_{k=0}^{n} m^{k}W_{m,r,q}(n, k)(a^{†})^{k}a^{k}  (16)

with initial conditions w_{m,r,q}(0, 0) = W_{m,r,q}(0, 0) = 1 and w_{m,r,q}(n, k) = W_{m,r,q}(n, k) = 0 for k > n and for k < 0, where the operators a^{†} and a satisfy the relation in (12).

Before proceeding we note that from (15) and (16),

w_{m,0,q}(n, k) = (−m)^{n−k}\left[{n \atop k}\right]_{q},  (17)
W_{m,0,q}(n, k) = m^{n−k}\left\{{n \atop k}\right\}_{q}.  (18)

Similarly, the r-Stirling numbers \begin{matrix} \\ [ & {n+r \atop k+r} & ] \end{matrix}_{r} and \begin{matrix} \\ \{ & {n+r \atop k+r} & \} \end{matrix}_{r} are specified by the horizontal generating functions

(x − r)_{n} = ∑_{k=0}^{n}(−1)^{n−k}\begin{matrix} \\ [ & {n + r \atop k + r} & ] \end{matrix}_{r}x^{k},

or, equivalently,

(x)_{n} = ∑_{k=0}^{n}(−1)^{n−k}\begin{matrix} \\ [ & {n + r \atop k + r} & ] \end{matrix}_{r}(x + r)^{k},

and

(x + r)^{n} = ∑_{k=0}^{n} \begin{matrix} \\ \{ & {n + r \atop k + r} & \} \end{matrix}_{r}(x)_{r}.

Hence, \begin{matrix} \\ [ & {n+r \atop k+r} & ] \end{matrix}_{q,r} and \begin{matrix} \\ \{ & {n+r \atop k+r} & \} \end{matrix}_{q,r}, the q-analogues of \begin{matrix} \\ [ & {n+r \atop k+r} & ] \end{matrix}_{r} and \begin{matrix} \\ \{ & {n+r \atop k+r} & \} \end{matrix}_{r}, respectively, are specified by the horizontal generating functions

(a^{†})^{n}a^{n} = ∑_{k=0}^{n}(−1)^{n−k}\begin{matrix} \\ [ & {n + r \atop k + r} & ] \end{matrix}_{q,r}(a^{†}a + r)^{k},  (19)

(a^{†}a + r)^{n} = ∑_{k=0}^{n} \begin{matrix} \\ \{ & {n + r \atop k + r} & \} \end{matrix}_{q,r}(a^{†})^{k}a^{k}.  (20)

It follows that

w_{1,r,q}(n, k) = (−1)^{n−k}\begin{matrix} \\ [ & {n + r \atop k + r} & ] \end{matrix}_{q,r},  (21)

W_{1,r,q}(n, k) = \begin{matrix} \\ \{ & {n + r \atop k + r} & \} \end{matrix}_{q,r}.  (22)

We will refer to the q-analogues in (19) and (20) as the (q, r)-Stirling numbers of the first and second kind, respectively.

Theorem 3. The (q, r)-Whitney numbers w_{m,r,q}(n, k) and W_{m,r,q}(n, k) satisfy the following identities:

w_{m,r,q}(n, k) = (−1)^{n−k} ∑_{i=k}^{n} \binom{i}{k}r^{i−k}m^{n−i}\left[{n \atop i}\right]_{q},  (23)

W_{m,r,q}(n, k) = ∑_{i=k}^{n} \binom{n}{i}r^{n−i}m^{i−k}\left\{{i \atop k}\right\}_{q}.  (24)

Proof. From Eq. (13), we get

m^{n}(a^{†})^{n}a^{n} = m^{n} ∑_{i=0}^{n}(−1)^{n−i}\left[{n \atop i}\right]_{q}(a^{†}a)^{i}

= m^{n} ∑_{i=0}^{n}(−1)^{n−i}\left[{n \atop i}\right]_{q} (\frac{ẑ − r}{m})^{i}

= m^{n} ∑_{i=0}^{n}(−1)^{n−i}\left[{n \atop i}\right]_{q} \frac{1}{m^{i}} ∑_{k=0}^{i} \binom{i}{k}ẑ^{k}(−r)^{i−k}

= ∑_{k=0}^{n}(−1)^{n−k} \{∑_{i=k}^{n} m^{n−i}\left[{n \atop i}\right]_{q}\binom{i}{k}r^{i−k}\} ẑ^{k},

where ẑ = ma^{†}a+r. Furthermore, comparing the coefficient of ẑ^{k} with that in equation (15) yields equation (23).

To prove equation (24), we write

(ma^{†}a + r)^{n} = ∑_{i=0}^{n} \binom{n}{i}r^{n−i}m^{i}(a^{†}a)^{i}

= ∑_{i=0}^{n} \binom{n}{i}r^{n−i}m^{i} ∑_{k=0}^{i} \left\{{i \atop k}\right\}_{q}(a^{†})^{k}a^{k}

= ∑_{k=0}^{n} \{∑_{i=k}^{n} r^{n−i}m^{i}\left\{{i \atop k}\right\}_{q}\binom{n}{i}\} (a^{†})^{k}a^{k}.

Comparing the coefficient of (a^{†})^{k}a^{k} with that in equation (16) gives us (24).

Remark 4. (a) As q → 1, we have

w_{m,r}(n, k) = ∑_{i=k}^{n}(−1)^{n−k}\binom{i}{k}r^{i−k}m^{n−i}\left[{n \atop i}\right];

W_{m,r}(n, k) = ∑_{i=k}^{n} \binom{n}{i}r^{n−i}m^{i−k}\left\{{i \atop k}\right\}.

(b) Note that of all the factors in equations (23) and (24) only the Stirling numbers are q-deformed.

The following corollary is a direct consequence of the previous theorem.

Corollary 5. The (q, r)-Stirling numbers are given by

\begin{matrix} \\ [ & {n + r \atop k + r} & ] \end{matrix}_{q,r} = ∑_{i=k}^{n} \binom{i}{k}r^{i−k}\left[{n \atop i}\right]_{q};  (25)

\begin{matrix} \\ \{ & {n + r \atop k + r} & \} \end{matrix}_{q,r} = ∑_{i=k}^{n} \binom{n}{i}r^{n−i}\left\{{i \atop k}\right\}_{q}.  (26)

3 Some recurrence relations

In this section, we present some recurrence relations involving the (q, r)-Whitney numbers. We recall the q-boson identities

[a, (a^{†})^{n}]_{q^{n}} = [n]_{q}(a^{†})^{n−1}

and

[a^{n}, a^{†}]_{q^{n}} = [n]_{q}a^{n−1},

that can be easily established by induction. The latter can also be written in the form

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

Theorem 6. The (q, r)-Whitney numbers w_{m,r,q}(n, k) and W_{m,r,q}(n, k) satisfy the following triangular recurrence relations:

w_{m,r,q}(n + 1, k) = q^{−n}(w_{m,r,q}(n, k − 1) − (m[n]_{q} + r)w_{m,r,q}(n, k)),  (27)
W_{m,r,q}(n + 1, k) = q^{k−1}W_{m,r,q}(n, k − 1) + (m[k]_{q} + r)W_{m,r,q}(n, k).  (28)

Proof. From equation (15), ∑_{k=0}^{n+1} w_{m,r,q}(n + 1, k)(ma^{†}a + r)^{k} = m^{n+1}(a^{†})^{n}(a^{†}a^{n})a

= m^{n+1}(a^{†})^{n}q^{−n}(a^{n}a^{†} − [n]_{q}a^{n−1})a
= m^{n+1}q^{−n} ((a^{†})^{n}a^{n}) (a^{†}a) − m^{n+1}q^{−n}[n]_{q}(a^{†})^{n}a^{n}
= q^{−n} ∑_{k=0}^{n} w_{m,r,q}(n, k)(ma^{†}a + r)^{k}(ma^{†}a + r − r) − mq^{−n}[n]_{q} ∑_{k=0}^{n} w_{m,r,q}(n, k)(ma^{†}a + r)^{k}

= q^{−n} ∑_{k=1}^{n+1} w_{m,r,q}(n, k − 1)(ma^{†}a + r)^{k} − q^{−n}(m[n]_{q} + r) ∑_{k=0}^{n} w_{m,r,q}(n, k)(ma^{†}a + r)^{k}

= q^{−n} ∑_{k=0}^{n+1} {w_{m,r,q}(n, k − 1) − (m[n]_{q} + r)w_{m,r,q}(n, k)} (ma^{†}a + r)^{k}.

Equating coefficients of (ma^{†}a + r)^{k} gives us (27) and a similar derivation yields equation (28).

Equations (27) and (28) are useful in computing the first few values of w_{m,r,q}(n, k) and W_{m,r,q}(n, k), using the initial values specified above. Remark 7. (a) From (27) we obtain the explicit expression

w_{m,r,q}(n, 0) = (−1)^{n}q^{− \frac{n(n−1)}{2}} ∏_{i=0}^{n−1}(m[i]_{q} + r).

On the other hand, the relation (23) yields

w_{m,r,q}(n, 0) = (−1)^{n} ∑_{i=0}^{n} r^{i}m^{n−i}\left[{n \atop i}\right]_{q}.

Equating these expressions and substituting x = \frac{r}{m} we obtain

∑_{i=0}^{n} \left[{n \atop i}\right]_{q}x^{i} = q^{− \frac{n(n−1)}{2}} ∏_{i=0}^{n−1}([i]_{q} + x).

This is a horizontal generating function for the q-Stirling numbers of the first kind in terms of a q-analogue of the rising factorial. Indeed, replacing x by −[s]_{q}, and noting that

[s]_{q} − [i]_{q} = q^{−i}[s − i]_{q}

and

∏_{i=0}^{n−1} q^{i} = q^{\binom{n}{2}},

we obtain

∑_{i=0}^{n} \left[{n \atop i}\right]_{q}(−1)^{i}[s]_{q}^{i} = (−1)^{n} ∏_{i=0}^{n−1}[s − i]_{q}.

(b) From (28) W_{m,r,q}(n + 1, 0) = rW_{m,r,q}(n, 0), hence W_{m,r,q}(n, 0) = r^{n}. The same result is obtained from (24). That is,

W_{m,r,q}(n, 0) = ∑_{i=0}^{n} \binom{n}{i}r^{n−i}m^{i}δ_{i,0} = r^{n}.

(c) As q → 1, we have

w_{m,r}(n + 1, k) = w_{m,r}(n, k − 1) − (mn + r)w_{m,r}(n, k);
W_{m,r}(n + 1, k) = W_{m,r}(n, k − 1) + (mk + r)W_{m,r}(n, k).

This confirms that w_{m,r,q}(n, k) and W_{m,r,q}(n, k) are proper q-analogues of w_{m,r}(n, k) and W_{m,r}(n, k), respectively.

As a consequence of the previous theorem, when m = 1 we have

Corollary 8. The (q, r)-Stirling numbers satisfy the following triangular recurrence relations:

\begin{matrix} \\ [ & {n + 1 + r \atop k + r} & ] \end{matrix}_{q,r} = q^{−n}\begin{matrix} \\ [ & {n + r \atop k − 1 + r} & ] \end{matrix}_{q,r} + ([n]_{q} + r)q^{−n}\begin{matrix} \\ [ & {n + r \atop k + r} & ] \end{matrix}_{q,r},

\begin{matrix} \\ \{ & {n + 1 + r \atop k + r} & \} \end{matrix}_{q,r} = q^{k−1}\begin{matrix} \\ \{ & {n + r \atop k − 1 + r} & \} \end{matrix}_{q,r} + ([k]_{q} + r)\begin{matrix} \\ \{ & {n + r \atop k + r} & \} \end{matrix}_{q,r}.

We can use these recurrence relations to compute the first few values of the (q, r)-Stirling numbers of the first and second kind, respectively.

Theorem 9. The (q, r)-Whitney numbers satisfy the following recurrence relations

w_{m,r+1,q}(n, ℓ) = ∑_{k=ℓ}^{n} \binom{k}{ℓ}(−1)^{k−ℓ}w_{m,r,q}(n, k),  (29)

W_{m,r+1,q}(n, k) = ∑_{ℓ=k}^{n} \binom{n}{ℓ}W_{m,r,q}(ℓ, k).  (30)

Proof. From equation (15), we have

m^{n}(a^{†})^{n}a^{n} = ∑_{k=0}^{n} w_{m,r,q}(n, k)(ma^{†}a + r)^{k}

= ∑_{k=0}^{n} w_{m,r,q}(n, k)((ma^{†}a + r + 1) − 1)^{k}

= ∑_{k=0}^{n} w_{m,r,q}(n, k) ∑_{ℓ=0}^{k} \binom{k}{ℓ}(−1)^{k−ℓ}(ma^{†}a + r + 1)^{ℓ}

= ∑_{ℓ=0}^{n}(ma^{†}a + r + 1)^{ℓ} ∑_{k=ℓ}^{n} \binom{k}{ℓ}(−1)^{k−ℓ}w_{m,r,q}(n, k).

On the other hand,

m^{n}(a^{†})^{n}a^{n} = ∑_{ℓ=0}^{n} w_{m,r+1,q}(n, ℓ)(ma^{†}a + r + 1)^{ℓ}.

Hence, by comparing the coefficients of (ma^{†}a + r + 1)^{ℓ} we obtain equation (29). Similarly, from equation (16)

(ma^{†}a + r + 1)^{n} = ∑_{k=0}^{n} m^{k}W_{m,r+1,q}(n, k)(a^{†})^{k}a^{k},

and since

(ma^{†}a + r + 1)^{n} = ∑_{ℓ=0}^{n} \binom{n}{ℓ}(ma^{†}a + r)^{ℓ}

= ∑_{ℓ=0}^{n} \binom{n}{ℓ} ∑_{k=0}^{ℓ} m^{k}W_{m,r,q}(ℓ, k)(a^{†})^{k}a^{k}

= ∑_{k=0}^{n} m^{k}(a^{†})^{k}a^{k} ∑_{ℓ=k}^{n} \binom{n}{ℓ}W_{m,r,q}(ℓ, k),

we obtain equation (30).

When m = 1, the theorem reduces to the recursion formulas for (q, r)-Stirling numbers. That is,

Corollary 10.

\begin{matrix} \\ [ & {n + r + 1 \atop l + r + 1} & ] \end{matrix}_{q,r+1} = ∑_{k=l}^{n}(−1)^{l−k}\binom{k}{l}\begin{matrix} \\ [ & {n + r \atop k + r} & ] \end{matrix}_{q,r},

\begin{matrix} \\ \{ & {n + r + 1 \atop k + r + 1} & \} \end{matrix}_{q,r+1} = ∑_{l=k}^{n} \binom{n}{l}\begin{matrix} \\ \{ & {l + r \atop k + r} & \} \end{matrix}_{q,r}.

4 Orthogonality and inverse relations

Theorem 11. The (q, r)-Whitney numbers w_{m,r,q}(n, k) and W_{m,r,q}(k, j) satisfy the following orthogonality relations:

∑_{k=j}^{n} W_{m,r,q}(n, k)w_{m,r,q}(k, j) = δ_{jn},  (31)

and

∑_{k=j}^{n} w_{m,r,q}(n, k)W_{m,r,q}(k, j) = δ_{jn},  (32)

where δ_{jn} is the Kronecker delta.

Proof. Using equation (15) we substitute m^{k}(a^{†})^{k}a^{k} in (16), obtaining

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

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

Comparing the coefficients of (ma^{†}a + r)^{j} yields equation (31). Equation (32) is obtained similarly.

The classical binomial inversion formula given by

f_{k} = ∑_{j=0}^{k} \binom{k}{j}g_{j} ⇔ g_{k} = ∑_{j=0}^{k}(−1)^{k−j}\binom{k}{j}f_{j}  (33)

can be a useful tool in deriving the explicit formula of the classical Stirling numbers of the second kind. The q-analogue of (33) is given by [8]

f_{k} = ∑_{j=0}^{k} \binom{k}{j}_{q}g_{j} ⇔ g_{k} = ∑_{j=0}^{k}(−1)^{k−j}q^{\binom{k−j}{2}}\binom{k}{j}_{q}f_{j},  (34)

The next theorem presents an inverse relation for the (q, r)-Whitney numbers w_{m,r,q}(n, k) and W_{m,r,q}(k, j).

Theorem 12. The (q, r)-Whitney numbers w_{m,r,q}(n, ℓ) and W_{m,r,q}(n, ℓ) satisfy the following inverse relation:

f_{n} = ∑_{ℓ=0}^{n} w_{m,r,q}(n, ℓ)g_{ℓ} ⇔ g_{n} = ∑_{ℓ=0}^{n} W_{m,r,q}(n, ℓ)f_{ℓ}.  (35)

Proof. By the hypothesis,

∑_{ℓ=0}^{n} W_{m,r,q}(n, ℓ)f_{ℓ} = ∑_{ℓ=0}^{n} W_{m,r,q}(n, ℓ) ∑_{k=0}^{ℓ} w_{m,r,q}(ℓ, k)g_{k}

= ∑_{k=0}^{n} {∑_{ℓ=k}^{n} W_{m,r,q}(n, ℓ)w_{m,r,q}(ℓ, k)} g_{k}

= ∑_{k=0}^{n} {δ_{kn}} g_{k}
= g_{n}.

The converse can be shown similarly.

The next theorem can be deduced in a similar way, from the orthogonality relations

Theorem 13. The (q, r)-Whitney numbers w_{m,r,q}(n, ℓ) and W_{m,r,q}(n, ℓ) satisfy the following inverse relation:

f_{ℓ} = ∑_{n=ℓ}^{∞} w_{m,r,q}(n, ℓ)g_{n} ⇔ g_{ℓ} = ∑_{n=ℓ}^{∞} W_{m,r,q}(n, ℓ)f_{n}.  (36)

5 (q, r)-Dowling polynomials and numbers

Cheon and Jung [7] defined the r-Dowling polynomials, denoted by D_{m,r}(n, x), in terms of sums of r-Whitney numbers of the second kind. That is,

D_{m,r}(n, x) = ∑_{k=0}^{n} W_{m,r}(n, k)x^{k}.  (37)

When x = 1, we obtain the r-Dowling numbers D_{m,r}(n) ≡ D_{m,r}(n, 1). The polynomials (37) are actually equivalent to the (r, β)-Bell polynomials G_{n,β,r}(x) of R. B. Corcino and C. B. Corcino [13]. That is,

D_{β,r}(n, x) = G_{n,β,r}(x).

Moreover,

• when m = 1 and r = 1, we recover the classical Dowling polynomials D(n, x) ≡ D_{1,1}(n, x);

• when m = 1 and r = 0, we recover the classical Bell polynomials B_{n}(x) ≡ D_{1,0}(n, x);

• when m = 1, we recover Mező’s [28] r-Bell polynomials B_{n,r}(x). That is, D_{1,r}(n, x) = B_{n,r}(x); and

• when m = α and r = 0, we recover the translated Dowling polynomials D̃_{(α)}(n; x) by Mangontarum et al. [25]. That is, D_{α,0}(n, x) = D̃_{(α)}(n; x).

Taking these into consideration, the next definition seems to be natural.

Definition 14. For non-negative integers n and k, and complex numbers m and r, the (q, r)-Dowling polynomials, denoted by D_{m,r,q}(n, x), are defined by

D_{m,r,q}(n, x) = ∑_{k=0}^{n} W_{m,r,q}(n, k)x^{k}  (38)

and the (q, r)-Dowling numbers, denoted by D_{m,r,q}(n), are defined by

D_{m,r,q}(n) = D_{m,r,q}(n, 1).  (39)

The coherent states

|γ⟩ = exp (−\frac{|γ|^{2}}{2} ) ∑_{k≥0} \frac{γ^{k}}{\sqrt{k!}}|k⟩,  (40)

where γ is an arbitrary (complex-valued) constant, satisfy a|γ⟩ = γ|γ⟩ and ⟨γ|γ⟩ = 1. Katriel [23] gave an illustration on how (40) can be a very useful tool in the derivation of certain Dobinski-type formulas. The q-coherent states corresponding to the q-Boson operators were defined as

|γ⟩_{q} = (ê_{q}(−|γ|^{2}))^{\frac{1}{2}} ∑_{k≥0} \frac{γ^{k}}{\sqrt{[k]_{q}!}}|k⟩  (41)

which satisfy a|γ⟩ = γ|γ⟩. Here, ê_{q}(x) is the type 2 q-exponential function given by

ê_{q}(x) = ∏_{i=1}^{∞}(1 + (1 − q)q^{i−1}x) = ∑_{i≥0} q^{\binom{i}{2}} \frac{x^{i}}{[i]_{q}!},  (42)

which is the inverse of the type 1 q-exponential function

e_{q}(x) = ∏_{i=1}^{∞}(1 − (1 − q)q^{i−1}x)^{−1} = ∑_{i≥0} \frac{x^{i}}{[i]_{q}!}.  (43)

That is, e_{q}(x)ê_{q}(−x) = 1.

Taking the expectation value of both sides of (16) with respect to |γ⟩ yields

⟨γ|(ma^{†}a + r)^{n}|γ⟩ = ∑_{k=0}^{n} m^{k}W_{m,r,q}(n, k)|γ|^{2k}.  (44)

The left-hand-side can be evaluated using the q-coherent states in (41), yielding

⟨γ|(ma^{†}a + r)^{n}|γ⟩ = ê_{q} (−|γ|^{2}) ∑_{k≥0} \frac{|γ|^{2k}}{[k]_{q}!} (m[k]_{q} + r)^{n}.  (45)

Defining x = m|γ|^{2} we obtain

∑_{k=0}^{n} W_{m,r,q}(n, k)x^{k} = ê_{q} (−\frac{x}{m}) ∑_{k≥0} (\frac{x}{m})^{k} \frac{(m[k]_{q} + r)^{n}}{[k]_{q}!} .  (46)

Using (38), the following theorem is easily observed.

Theorem 15. The (q, r)-Dowling polynomials D_{m,r,q}(n, x) and the (q, r)-Dowling numbers D_{m,r,q}(n) have the following explicit formulas:

D_{m,r,q}(n, x) = ê_{q} (−\frac{x}{m}) ∑_{k≥0} (\frac{x}{m})^{k} \frac{(m[k]_{q} + r)^{n}}{[k]_{q}!} ,  (47)

and

D_{m,r,q}(n) = ê_{q} (−m^{−1}) ∑_{k≥0} \frac{(m[k]_{q} + r)^{n}}{m^{k}[k]_{q}!} .  (48)

Proof. (48) can be obtained by letting x = 1 in (47).

Katriel [23] defined the q-Bell polynomial as

∑_{ℓ=0}^{k} \left\{{k \atop ℓ}\right\}_{q}x^{ℓ} = ê_{q}(x) ∑_{m=1}^{∞} x^{m} \frac{[m]_{q}^{k}}{[m]_{q}!}.  (49)

Expanding the right-hand side using (42) yields

∑_{ℓ=0}^{k} \left\{{k \atop ℓ}\right\}_{q}x^{ℓ} = ∑_{ℓ=0}^{∞} \frac{x^{ℓ}}{[ℓ]_{q}!} ∑_{j=0}^{ℓ}(−1)^{ℓ−j}q^{\binom{ℓ−j}{2}}\binom{ℓ}{j}_{q}[j]_{q}^{k}.  (50)

Equating coefficients of equal powers of x gives us

\left\{{k \atop ℓ}\right\}_{q} = \frac{1}{[ℓ]_{q}!} ∑_{j=0}^{ℓ}(−1)^{ℓ−j}q^{\binom{ℓ−j}{2}}\binom{ℓ}{j}_{q}[j]_{q}^{k}.  (51)

Notice that as q → 1, (51) reduces to the well-known explicit formula of \left\{{k \atop j}\right\}. That is

lim_{q→1} \left\{{k \atop ℓ}\right\}_{q} = \frac{1}{ℓ!} ∑_{j=0}^{ℓ}(−1)^{ℓ−j}\binom{ℓ}{j}j^{k}.  (52)

In the following theorem, we will present an expression analogous to (51) for the q-analogue W_{m,r,q}(n, k).

Theorem 16. The (q, r)-Whitney numbers of the second kind, W_{m,r,q}(n, k), have the following explicit formula:

W_{m,r,q}(n, ℓ) = \frac{1}{m^{ℓ}[ℓ]_{q}!} ∑_{k=0}^{ℓ}(−1)^{ℓ−k}q^{\binom{ℓ−k}{2}}\binom{ℓ}{k}_{q}(m[k]_{q} + r)^{n}.  (53)

Proof. Substituting y = \frac{x}{m} in (47) gives us

∑_{k=0}^{n} m^{k}W_{m,r,q}(n, k)y^{k} = ∑_{i≥0} q^{\binom{i}{2}} \frac{(−y)^{i}}{[i]_{q}!} ∑_{k≥0} y^{k} \frac{(m[k]_{q} + r)^{n}}{[k]_{q}!}

= ∑_{ℓ≥0} \frac{y^{ℓ}}{[ℓ]_{q}!} ∑_{k=0}^{ℓ}(−1)^{ℓ−k}q^{\binom{ℓ−k}{2}}\binom{ℓ}{k}_{q}(m[k]_{q} + r)^{n}.

Equating the coefficients of equal powers of y on both sides of this equation we obtain equation (53).

Note that as q → 1, we have

lim_{q→1} W_{m,r,q}(n, ℓ) = \frac{1}{m^{ℓ}ℓ!} ∑_{k=0}^{ℓ}(−1)^{ℓ−k}\binom{ℓ}{k}(mk + r)^{n}
= W_{m,r}(n, ℓ).

Furthermore,

lim_{q→1} W_{m,1,q}(n, l) = W_{m}(n, l).

Remark 17. We can also prove (53) in the following manner: First, we write (16) as

(m[ℓ]_{q} + r)^{n} = ∑_{k=0}^{n} m^{k}W_{m,r,q}(n, k)[ℓ]_{q,k}
= ∑_{k=0}^{ℓ} \binom{ℓ}{k}_{q} \{\frac{m^{k}W_{m,r,q}(n, k)[ℓ]_{q,k}}{\binom{ℓ}{k}q} \} .

Next, we apply the q-binomial inversion formula in (34) which gives us

\frac{m^{ℓ}W_{m,r,q}(n, ℓ)[ℓ]_{q,ℓ}}{\binom{k}{k}q} = ∑_{k=0}^{ℓ}(−1)^{ℓ−k}q^{\binom{ℓ−k}{2}}\binom{l}{k}_{q}(m[k]_{q} + r)^{n}.

This is precisely the explicit formula obtained in the previous theorem.

Now, using (53),

∑_{n≥0} W_{m,r,q}(n, k)\frac{t^{n}}{[n]_{q}!} = ∑_{n≥0} ∑_{j=0}^{k} \frac{(−1)^{k−j}}{m^{k}[k]_{q}!} q^{\binom{k−j}{2}}\binom{k}{j}_{q}(m[j]_{q} + r)^{n} \frac{t^{n}}{[n]_{q}!}
= \frac{1}{m^{k}[k]_{q}!} ∑_{j=0}^{k}(−1)^{k−j}q^{\binom{k−j}{2}}\binom{k}{j}_{q}e_{q} [(m[j]_{q} + r)t] ,

where e_{q}(x) is the type 1 q-exponential function in (43). Thus, we have the following theorem. Theorem 18. The (q, r)-Whitney numbers of the second kind satisfy the following exponential generating function:

∑_{n≥0} W_{m,r,q}(n, k)\frac{t^{n}}{[n]_{q}!} = \frac{1}{m^{k}[k]_{q}!} ∑_{j=0}^{n}(−1)^{k−j}q^{\binom{k−j}{2}}\binom{k}{j}_{q}e_{q} [(m[j]_{q} + r)t] .  (54)

Remark 19. As q → 1, we have

lim_{q→1} ∑_{n≥0} W_{m,r,q}(n, k)\frac{t^{n}}{[n]_{q}!} = \frac{e^{rt}}{k!} (\frac{e^{mt} − 1}{m})^{k} ,

which is the exponential generating function of the r-Whitney numbers of the second kind.

The q-difference operator [24] can be written in the form

Δ_{q}^{k}f(x) = ∑_{j=0}^{k}(−1)^{k−j}q^{\binom{k−j}{2}}\binom{k}{j}_{q}f(x + j).  (55)

We are now ready to state the next theorem. Theorem 20. The (q, r)-Whitney numbers of the second kind satisfy the following identity:

∑_{n≥0} W_{m,r,q}(n, k)\frac{t^{n}}{[n]_{q}!} = \{Δ_{q}^{k} (\frac{e_{q}[(m[x]_{q} + r)t]}{m^{k}[k]_{q}!})\}_{x=0} .  (56)

Proof. (56) follows directly from (54) and (55).

The next corollary is easily verified. Corollary 21. The (q, r)-Whitney numbers of the second kind can be expressed explicitly as

W_{m,r,q}(n, k) = \{Δ_{q}^{k} (\frac{(m[x]_{q} + r)^{n}}{m^{k}[k]_{q}!} )\}_{x=0} .  (57)

6 Further identities for the (q, r)-Whitney numbers

Graham et al. [21] presented a useful set of Stirling number identities while Katriel [22] presented the q-analogues of all but two of them. Three of these identities are generalized to the (q, r)-Whitney numbers using appropriate modifications of the procedures presented by Katriel [22]. Their derivation requires the following. Lemma 22. For f(x) a polynomial, the operator identity

a^{†}f(1 + qa^{†}a)a = a^{†}af(a^{†}a),  (58)

holds. Proof. We write the q-commutation relation, equation (12), in the form aa^{†} = 1 + qa^{†}a. It follows that

(a^{†}a)(a^{†}a)^{k} = a^{†}(aa^{†})^{k}a = a^{†}(1 + qa^{†}a)^{k}a.

For f(x) = ∑_{k} c_{k}x^{k} we obtain

a^{†}af(a^{†}a) = ∑_{k} c_{k}(a^{†}a)(a^{†}a)^{k}
= ∑_{k} c_{k}a^{†}(1 + qa^{†}a)^{k}a = a^{†} (∑_{k} c_{k}(1 + qa^{†}a)^{k}) a
= a^{†}f(1 + qa^{†}a)a.

Remark 23. The lemma can also be written in the form

a^{†}g(a^{†}a)a = a^{†}ag (\frac{1}{q}(a^{†}a − 1)) ,  (59)

where g(x) is a polynomial. Theorem 24 (Identity 1). The (q, r)-Whitney numbers of the second kind satisfy

W_{m,r,q}(n + 1, k) − rW_{m,r,q}(n, k) = ∑_{ℓ=k−1}^{n} \binom{n}{ℓ}q^{ℓ}(m + r(1 − q))^{n−ℓ}W_{m,r,q}(ℓ, k − 1).

Proof. In terms of the identity (58) and with the aid of (16)

a^{†}(m(1 + qa^{†}a) + r)^{n}a = a^{†}a(ma^{†}a + r)^{n}
= \frac{1}{m}(ma^{†}a + r − r)(ma^{†}a + r)^{n}
= \frac{1}{m}(ma^{†}a + r)^{n+1} − \frac{r}{m}(ma^{†}a + r)^{n}
= ∑_{k=0}^{n+1} m^{k−1}(a^{†})^{k}a^{k}(W_{m,r,q}(n + 1, k) − rW_{m,r,q}(n, k)).

On the other hand, defining α = m + r(1 − q) (which will hold throught the present section),

a^{†}(m(1 + qa^{†}a) + r)^{n}a = a^{†}(q(ma^{†}a + r) + α)^{n}a
= a^{†} (∑_{ℓ=0}^{n} \binom{n}{ℓ}q^{ℓ}α^{n−ℓ}(ma^{†}a + r)^{ℓ}) a
= ∑_{ℓ=0}^{n} \binom{n}{ℓ}q^{ℓ}α^{n−ℓ} ∑_{k=0}^{ℓ} m^{k}W_{m,r,q}(ℓ, k)(a^{†})^{k+1}a^{k+1}
= ∑_{k=1}^{n+1} m^{k−1}(a^{†})^{k}a^{k} ∑_{ℓ=k−1}^{n} \binom{n}{ℓ}q^{ℓ}α^{n−ℓ}W_{m,r,q}(ℓ, k − 1).

Equating coefficients of m^{k−1}(a^{†})^{k}a^{k} the theorem follows.

For r = 0 this identity reduces to the q-Stirling numbers identity [22, identity 1]

W_{m,0,q}(n + 1, k) = ∑_{ℓ=k−1}^{n} \binom{n}{ℓ}q^{ℓ}m^{n−ℓ}W_{m,0,q}(ℓ, k − 1).

The following corollary is an immediate consequence of the previous theorem.

Corollary 25. As q → 1,

W_{m,r}(n + 1, k) − rW_{m,r}(n, k) = ∑_{ℓ=k−1}^{n} \binom{n}{ℓ}m^{n−ℓ}W_{m,r}(ℓ, k − 1).

Theorem 26 (Identity 2). The (q, r)-Whitney numbers of the first kind satisfy

w_{m,r,q}(n + 1, ℓ) = ∑_{k=ℓ−1}^{n} \frac{1}{q^{k}} w_{m,r,q}(n, k)( − (m + r(1 − q))^{k−ℓ}

· (\binom{k}{ℓ − 1}(−(m + r(1 − q))) − r\binom{k}{ℓ}) .

Proof. We note that from (15),

m^{n+1}(a^{†})^{n+1}a^{n+1} = ∑_{ℓ=0}^{n+1} w_{m,r,q}(n + 1, ℓ)(ma^{†}a + r)^{ℓ}.

On the other hand, using (59),

m^{n+1}(a^{†})^{n+1}a^{n+1} = ma^{†}(m^{n}(a^{†})^{n}a^{n})a
= ma^{†}( ∑_{k=0}^{n} w_{m,r,q}(n, k)(ma^{†}a + r)^{k})a
= ma^{†}a ∑_{k=0}^{n} w_{m,r,q}(n, k)(\frac{m}{q}(a^{†}a − 1) + r)^{k}
= ma^{†}a ∑_{k=0}^{n} w_{m,r,q}(n, k)\frac{1}{q^{k}} ((ma^{†}a + r) − α)^{k}
= ((ma^{†}a + r) − r) ∑_{k=0}^{n} w_{m,r,q}(n, k)\frac{1}{q^{k}} ∑_{ℓ=0}^{k} \binom{k}{ℓ}(ma^{†}a + r)^{ℓ}(−α)^{k−ℓ}
= ∑_{k=0}^{n} w_{m,r,q}(n, k)\frac{1}{q^{k}} ∑_{ℓ=0}^{k} \binom{k}{ℓ}(ma^{†}a + r)^{ℓ+1}(−α)^{k−ℓ}

−r ∑_{k=0}^{n} w_{m,r,q}(n, k)\frac{1}{q^{k}} ∑_{ℓ=0}^{n} \binom{k}{ℓ}(ma^{†}a + r)^{ℓ}(−α)^{k−ℓ}

= ∑_{ℓ=0}^{n+1}(ma^{†}a + r)^{ℓ} ∑_{k=ℓ−1}^{n} \frac{1}{q^{k}} w_{m,r,q}(n, k)(−α)^{k−ℓ} ·

· (\binom{k}{ℓ − 1}(−α) − r\binom{k}{ℓ}) .

Equating the coefficients of equal powers of ma^{†}a + r we obtain the theorem.

For r = 0, we recover the q-Stirling numbers identity [22, identity 2]

w_{m,0,q}(n + 1, ℓ) = ∑_{k=ℓ−1}^{n} \frac{1}{q^{k}} w_{m,0,q}(n, k)(−m)^{k−ℓ+1}\binom{k}{ℓ − 1},

Moreover, we have the following corollary: Corollary 27. As q → 1,

w_{m,r}(n + 1, ℓ) = − ∑_{k=ℓ−1}^{n} w_{m,r}(n, k)(−m)^{k−ℓ} (m\binom{k}{ℓ − 1} + r\binom{k}{ℓ}) .

Theorem 28 (Identity 3). The (q, r)-Whitney numbers of the second kind satisfy

W_{m,r,q}(n, k−1) = \frac{1}{q^{n}} ∑_{ℓ=k}^{n+1}(−m−r(1−q))^{n−ℓ} (\binom{n}{ℓ − 1}(−m − r(1 − q)) − \binom{n}{ℓ}r) W_{m,r,q}(ℓ, k).

Proof. Note that

a^{†}(ma^{†}a + r)^{n}a = ∑_{k=0}^{n} m^{k}W_{m,r,q}(n, k)(a^{†})^{k+1}a^{k+1}
= ∑_{k=1}^{n+1} m^{k−1}W_{m,r,q}(n, k − 1)(a^{†})^{k}a^{k},

and on the other hand, using (59),

a^{†}(ma^{†}a + r)^{n}a = a^{†}a(\frac{m}{q}(a^{†}a − 1) + r)^{n} = a^{†}a\frac{1}{q^{n}} (ma^{(†}a + r) − α)^{n}
= \frac{1}{m}((ma^{†}a + r) − r)\frac{1}{q^{n}} ∑_{ℓ=0}^{n} \binom{n}{ℓ}(ma^{†}a + r)^{ℓ}(−α))^{n−ℓ}
= \frac{1}{mq^{n}} ∑_{ℓ=1}^{n+1}(ma^{†}a + r)^{ℓ}(−α)^{n−ℓ} · (\binom{n}{ℓ − 1}(−α) − r\binom{n}{ℓ})
= \frac{1}{mq^{n}} ∑_{k=0}^{n+1} m^{k}(a^{†})^{k}a^{k} ∑_{ℓ=k}^{n+1}(−α)^{n−ℓ} (\binom{n}{ℓ − 1}(−α) − \binom{n}{ℓ}r) W_{m,r,q}(ℓ, k).

Equating the coefficients of (a^{†})^{k}a^{k} we obtain the theorem.

For r = 0 this theorem reduces to

W_{m,0,q}(n, k − 1) = \frac{1}{q^{n}} ∑_{ℓ=k}^{n+1}(−m)^{n+1−ℓ}\binom{n}{ℓ − 1}W_{m,0,q}(ℓ, k).

Using equation (18), we can verify that this is just the q-Stirling numbers identity [22, identity 3]. The next corollary is easily verified.

Corollary 29. As q → 1,

W_{m,r}(n, k − 1) = ∑_{ℓ=k}^{n+1}(−m)^{n−ℓ} [\binom{n}{ℓ − 1}(−m) − \binom{n}{ℓ}r] W_{m,r}(ℓ, k).

Presently, much is yet to be learnt regarding the (q, r)-Whitney numbers. The classical r-Whitney and Stirling numbers are known to have various applications in different fields. It is tempting to explore applications for the (q, r)-Whitney numbers.

To close this section, Corcino and Hererra [17] defined the q-analogue of the limit of the differences of the generalized factorial F_{α,γ}(n, k) in (5), denoted by ϕ_{α,γ}[n, k]_{q}. ϕ_{α,γ}[n, k]_{q} can be defined in terms of the relation

∑_{k=0}^{n} ϕ_{α,γ}[n, k]_{q}t^{k} = ⟨t + [γ]_{q}|[α]_{q}⟩_{n}^{q} ,  (60)

where

⟨t + [γ]_{q}|[α]_{q}⟩_{n}^{q} = ∏_{j=0}^{n−1} (t + [γ]_{q} − [jα]_{q}) .  (61)

The numbers ϕ_{α,γ}[n, k]_{q} are actually q-analogues of the numbers w_{m,r}(n, k). Similarly, Corcino and Montero [18] defined the q-analogue σ[n, k]_{q}^{β,r} of the Rucinski-Voigt numbers in terms of the reccurence relation

σ[n, k]_{q}^{β,r} = σ[n − 1, k − 1]_{q}^{β,r} + ([kβ]_{q} + [r]_{q}) σ[n − 1, k]_{q}^{β,r}.  (62)

σ[n, k]_{q}^{β,r} is also a q-analogue of the numbers \left⟨{n \atop k}\right⟩_{r,β} and W_{m,r}(n, k). However, by comparing the defining relations for ϕ_{α,γ}[n, k]_{q} and σ[n, k]_{q}^{β,r} with those of the(q, r)-Whitney numbers w_{m,r,q}(n, k) and W_{m,r,q}(n, k), respectively, we note that they represent distinctly motivated q-analogues that cannot be simply related to one another.

7 Acknowledgments

The authors would like to thank the editor-in-chief for his helpful comments and suggestions, and the referee for carefully reading the manuscript and giving invaluable recommendations which helped improve this paper.

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2010 Mathematics Subject Classification: Primary 11B83; Secondary 11B73, 05A30. Keywords: Whitney number, Stirling number, Boson operator, q-analogue.

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

Received May 13 2015; revised version received, September 1 2015. Published in Journal of Integer Sequences, September 7 2015.

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