Some Theorems and Applications of the (q, r)-Whitney Numbers

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

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

Abstract

The (q, r)-Whitney numbers were recently defined in terms of the q-Boson operators, and several combinatorial properties which appear to be q-analogues of similar properties were studied. In this paper, we obtain elementary and complete symmetric polynomial forms for the (q, r)-Whitney numbers, and give combinatorial interpretations in the context of A-tableaux. We also obtain convolution-type identities using the combinatorics of A-tableaux. Lastly, we present applications and theorems related to discrete q-distributions.

1 Introduction

In a recent paper, the author and Katriel [21] introduced a new approach to generate qanalogues of Stirling and Whitney-type numbers. In this paper, the (q, r)-Whitney numbers of the first and second kinds were defined as coefficients in

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

and

(ma^{†}a + r)^{n} = ∑_{k=0}^{n} m^{k}W_{m,r,q}(n, k)(a^{†})^{k}a^{k},  (2)

respectively (cf. [21]), by using as framework, the q-Boson operators a^{†} and a of Arik and Coon [2] which satisfy the commutation relation

[a, a^{†}]_{q} ≡ aa^{†} − qa^{†}a = 1.  (3)

By convention, 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. Several combinatorial properties were already established, including the following triangular recurrence relations [21, Theorem 6]:

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)),  (4)

with [n]_{q} = \frac{q^{n}−1}{q−1}, the q-integer, and

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).  (5)

From here, one readily obtains

w_{m,r,q}(n, 0) = (−1)^{n}q^{−\binom{n}{2}} ∏_{i=0}^{n−1}(m[i]_{q} + r),  (6)

w_{m,r,q}(n, n) = q^{−\binom{n}{2}},  (7)
W_{m,r,q}(n, 0) = r^{n},  (8)

and

W_{m,r,q}(n, n) = q^{\binom{n}{2}}.  (9)

The identities presented in Eqs. (4) and (5) can be used as tools to obtain further combinatorial identities for w_{m,r,q}(n, k) and W_{m,r,q}(n, k). For instance, with the aid of these recurrence relations, the vertical recurrence relations

w_{m,r,q}(n + 1, k + 1) = ∑_{j=k}^{n}(−1)^{n−j}q^{\binom{j}{2}−\binom{n+1}{2}}w_{m,r,q}(j, k) ∏_{i=j+1}^{n}(m[i]_{q} + r),  (10)

with ∏_{i=j+1}^{n}(m[i]_{q} + r) = 1 when j = n, and

W_{m,r,q}(n + 1, k + 1) = q^{k} ∑_{j=k}^{n}(m[k + 1]_{q} + r)^{n−j}W_{m,r,q}(j, k),  (11)

can be proved by induction, as well as the rational generating function of the (q, r)-Whitney numbers of the second kind given by

∑_{n=k}^{∞} W_{m,r,q}(n, k)t^{n} = \frac{q^{\binom{k}{2}}t^{k}}{∏_{i=0}^{k} (1 − (m[i]_{q} + r)t)}.  (12)

On the other hand, the horizontal recurrence relations

w_{m,r,q}(n, k) = q^{n} ∑_{j=0}^{n−k}(m[n]_{q} + r)^{j}w_{m,r,q}(n + 1, k + j + 1)  (13)

and

W_{m,r,q}(n, k) = ∑_{j=0}^{n−k}(−1)^{j}q^{\binom{k}{2}−\binom{k+j+1}{2}} \frac{∏_{i=0}^{k+j}(m[i]_{q} + r)}{∏_{i=0}^{k}(m[i]_{q} + r)}W_{m,r,q}(n + 1, k + j + 1) (14)

can be verified by evaluating the right-hand sides using Eqs. (4) and (5). Before proceeding, we note that Eqs. (10) and (11) follow a behaviour similar to that of the Chu-Shi-Chieh’s identity (see [6]) for the classical binomial coefficients given by

\binom{n + 1}{k + 1} = \binom{k}{k} + \binom{k + 1}{k} + ⋯ + \binom{n}{k},

while Eqs. (13) and (14) are analogous with

\binom{n}{k} = \binom{n + 1}{k + 1} − \binom{n + 1}{k + 2} + ⋯ + (−1)^{n−k}\binom{n + 1}{n + 1},

another known identity for the classical binomial coefficients.

The purpose of this paper is to express the (q, r)-Whitney numbers of both kinds in symmetric polynomial forms. This proves to be useful in establishing combinatorial interpretations in terms of A-tableaux. In return, remarkable convolution-type identities are obtained and several other interesting theorems are also presented.

2 Explicit formulas in symmetric polynomial forms

2.1 (q, r)-Whitney numbers of the first kind

Expanding the falling factorial (x)_{n} = x(x − 1) ⋯ (x − n + 1) in powers of x, we obtain

(x)_{n} = ∑_{k=0}^{n}(−1)^{n−k}x^{k} ∑_{1≤i_{1}<i_{2}<···<i_{n−k}≤n−1} ∏_{j=1}^{n−k} i_{j} ,

which yield the well-known expression for the Stirling numbers of the first kind in terms of elementary symmetric functions. This relation can be generalized to the q-Stirling numbers as follows:

[x]_{q}[x − 1]_{q} ⋯ [x − n + 1]_{q} = [x]_{q}([x]_{q} + q^{x})([x]_{q} + q^{x}[2]_{q}) ⋯ ([x]_{q} + q^{x}[n − 1]_{q})
= ∑_{k=0}^{n}[x]_{q}^{k} · q^{x(n−k)} ∑_{1≤i_{1}<i_{2}<···<i_{n−k}≤n−1} ∏_{j=1}^{n−k}[i_{j}]_{q} .

To further generalize this procedure to the (q, r)-Whitney numbers of the first kind, recall that application of both sides of the defining relation in Eq. (1) on the q-boson number state |ℓ⟩ gives

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

Since both sides of this relation are finite polynomials in ℓ, and since the relation is valid for all integer ℓ, it remains valid when ℓ is replaced by the real number x, i.e.,

m^{n}[x]_{q}[x − 1]_{q} ⋯ [x − n + 1]_{q} = ∑_{k=0}^{n} w_{m,r,q}(n, k)(m[x]_{q} + r)^{k} .  (15)

Now, defining y = [x]_{q} + α, where α = \frac{r}{m}, we note that [x − i]_{q} = q^{−i}(y − α − [i]_{q}). Hence,

m^{n}[x]_{q}[x−1]_{q} ⋯ [x−n+1]_{q} = ∑_{k=0}^{n}(m[x]_{q}+r)^{k}q^{−\binom{n}{2}}(−1)^{n−k} ∑_{0≤i_{1}<i_{2}<···<i_{n−k}≤n−1} ∏_{j=1}^{n−k}(m[i_{j}]_{q}+r) .

(16)

The identity in the next theorem is obtained by comparing the right-hand-sides of Eqs. (15) and (16).

Theorem 1. The (q, r)-Whitney numbers of the first kind satisfy the following explicit form

w_{m,r,q}(n, k) = (−1)^{n−k}q^{−\binom{n}{2}} ∑_{0≤i_{1}<i_{2}<···<i_{n−k}≤n−1} ∏_{j=1}^{n−k}(r + [i_{j}]_{q}m).  (17)

Remark 2. The sum within this theorem is the symmetric polynomial of degree n − k in the n variables {(r + [i]_{q}m); i = 0, 1, …, n − 1}. For r = 0 all the terms with i_{1} = 0 vanish so the summation starts at 1, which is consistent with the expressions presented above for the Stirling and q-Stirling numbers of the first kind.

The above theorem can also be proved by induction as follows:

Alternative proof of Theorem 1. The theorem readily yields w_{m,r,q}(0, 0) = 1. Making the induction hypothesis that the theorem is true up to n, for all k = 0, 1, …, n, we prove it for n + 1 and k = 0, 1, …, n, via the recurrence relation (4). Thus,

w_{m,r,q}(n + 1, k) = q^{−n}((−1)^{n+1−k}q^{−\binom{n}{2}} ∑_{0≤i_{1}<i_{2}<···<i_{n+1−k}≤n−1} ∏_{j=1}^{n+1−k}(r + [i_{j}]_{q}m)

−(m[n]_{q} + r)(−1)^{n−k}q^{−\binom{n}{2}} ∑_{0≤i_{1}<i_{2}<···<i_{n−k}≤n−1} ∏_{j=1}^{n−k}(r + [i_{j}]_{q}m))

= q^{−\binom{n+1}{2}}(−1)^{n+1−k}q^{−\binom{n}{2}}( ∑_{0≤i_{1}<i_{2}<···<i_{n+1−k}≤n−1} ∏_{j=1}^{n+1−k}(r + [i_{j}]_{q}m)

+(m[n]_{q} + r) ∑_{0≤i_{1}<i_{2}<···<i_{n−k}≤n−1} ∏_{j=1}^{n−k}(r + [i_{j}]_{q}m))

The first term within the large paretheses contains all products of n + 2 − k distinct factors out of {(r + [i]_{q}m); i = 0, 1, …, n − 1}, whereas the second term contains all products of n + 2 − k distinct factors, one of which is (r + m[n]_{q}) and the others chosen out of {(r + [i]_{q}m); i = 0, 1, …, n − 1}. Together, these sums yield

∑_{0≤i_{1}<i_{2}<···<i_{n+1−k}≤n} ∏_{j=0}^{n+1−k}(r + [i_{j}]_{q}m),

thus establishing the theorem for the range of indices specified above. Finally, the theorem yields w_{m,r,q}(n + 1, n + 1) = q^{−\binom{n+1}{2}}, in agreement with (7).

As q → 1, the explicit formula (17) reduces to an expression for the r-Whitney numbers of the first kind given by

w_{m,r}(n, k) = (−1)^{n−k} ∑_{0≤i_{1}<i_{2}<···<i_{n−k}≤n−1} ∏_{j=1}^{n−k}(r + i_{j}m).  (18)

An equivalent of this identity was reported by Mangontarum et al. [18, Theorem 6]. For m = 1 and r = 0, (17) reduces to an explicit formula for a q-analogue of the Stirling numbers of the first kind, viz,

\left[{n \atop k}\right]_{q} = (−1)^{n−k}q^{−\binom{n}{2}} ∑_{0≤i_{1}<i_{2}<···<i_{n−k}≤n−1} ∏_{j=1}^{n−k}[i_{j}]_{q},  (19)

where \left[{n \atop k}\right]_{q} denote the q-Stirling numbers of the first kind defined by

[x]_{q,n} = ∑_{k=1}^{n}(−1)^{n−k}\left[{n \atop k}\right]_{q}[x]_{q}^{k},  (20)

[x]_{q,n} = [x]_{q}[x − 1]_{q}[x − 2]_{q} ⋯ [x − n + 1]_{q} (cf. [4]). For any given set of n − k integers that satisfy 1 < i_{2} < ⋯ < i_{n−k} < n − 1, let

{ℓ_{1}, ℓ_{2}, …, ℓ_{k}} ≡ {1, 2, 3, …, n − 1} − {i_{1}, i_{2}, …, i_{n−k}}

be the complement with respect to {1, 2, 3, …, n − 1}. It follows that

∏_{j=0}^{n−k}[i_{j}]_{q} = \frac{[n − 1]_{q}!}{∏_{j=0}^{k}[ℓ_{j}]_{q}} .  (21)

This allows (19) to be written in the form

\left[{n \atop k}\right]_{q} = q^{−\binom{n}{2}}[n − 1]_{q}! ∑_{0≤i_{1}<i_{2}<···<i_{n−k}≤n−1} \frac{1}{∏_{j=0}^{k}[ℓ_{j}]_{q}} .  (22)

As q → 1, one recovers from (19) Comtet’s [8] identity given by

\left[{n \atop k}\right] = (−1)^{n−k} ∑_{0≤i_{1}<i_{2}<···<i_{n−k}≤n−1} ∏_{j=1}^{n−k} i_{j},  (23)

while (22) yields Adamchik’s [1] identity for the Stirling numbers of the first kind given by

\left[{n \atop k}\right] = (n − 1)! ∑_{0≤i_{1}<i_{2}<···<i_{n−k}≤n−1} \frac{1}{∏_{j=0}^{k} ℓ_{j}} .  (24)

2.2 (q, r)-Whitney numbers of the second kind

Theorem 3. The (q, r)-Whitney numbers of the second kind satisfy the following explicit form:

W_{m,r,q}(n, k) = q^{\binom{k}{2}} ∑_{c_{0}+c_{1}+···+c_{k}=n−k} ∏_{j=0}^{k}(m[j]_{q} + r)^{c_{j}},  (25)

where c_{0}, c_{1}, …, c_{k} are non-negative integers.

Proof. We proceed by induction over n. First, we note that the theorem is satisfied when n = k = 0. That is, W_{m,r,q}(0, 0) = 1. Making the induction hypothesis that the theorem holds up to n (for all k = 0, 1, …, n) we show, using the recurrence relation (5), that it holds for n + 1 and k = 0, 1, …, n. Thus,

W_{m,r,q}(n + 1, k) = q^{k−1}q^{\binom{k−1}{2}} ∑_{c_{0}+c_{1}+···+c_{k−1}=n+1−k} ∏_{j=0}^{k−1}(m[j]_{q} + r)^{c_{j}}

+(m[k]_{q} + r)q^{\binom{k}{2}} ∑_{c_{0}+c_{1}+···+c_{k}=n−k} ∏_{j=0}^{k}(m[j]_{q} + r)^{c_{j}}

= q^{\binom{k}{2}}( ∑_{c_{0}+c_{1}+···+c_{k−1}=n+1−k} ∏_{j=0}^{k−1}(m[j]_{q} + r)^{c_{j}}

+(m[k]_{q} + r) ∑_{c_{0}+c_{1}+···+c_{k}=n−k} ∏_{j=0}^{k}(m[j]_{q} + r)^{c_{j}})

Now, the first term within the big paretheses is a sum of products of n + 1 − k factors, non of which contains (m[k]_{q} + r). The second term is again a sum of n + 1 − k factors, each one of which containing (m[k]_{q} + r) at least once. Thus,

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

To complete the proof we need to show that the theorem holds for n + 1 and k = n + 1. For this case the theorem yields W_{m,r,q}(n+1, n+1) = q^{\binom{n+1}{2}}, which is in agreement with (9).

Apart from q^{\binom{k}{2}}, (25) is a homogeneous complete symmetric polynomial of degree n − k in the variables {(r + [j]_{q}m); j = 0, 1, 2, …, k}. As q → 1, we obtain

W_{m,r}(n, k) = ∑_{c_{0}+c_{1}+···+c_{k}=n−k} ∏_{j=0}^{k}(r + mj)^{c_{j}},  (26)

and for r = 0, (25) reduces to an expression for the q-Stirling numbers of the second kind, viz,

\left\{{n \atop k}\right\}_{q} = q^{\binom{k}{2}} ∑_{c_{0}+c_{1}+···+c_{k}=n−k}[1]_{q}^{i_{1}}[2]_{q}^{i_{2}} ⋯ [k]_{q}^{i_{k}}.  (27)

The q-Stirling numbers of the second kind were originally defined as

[x]_{q}^{n} = ∑_{k=1}^{n} \left\{{n \atop k}\right\}_{q}[x]_{q,k}  (28)

(cf. [4]). Moreover, when q → 1, Eq. (27) yields an expression for the classical Stirling numbers of the second kind reported by Comtet [8].

Notice that from the inner product

∏_{j=0}^{k}(m[j]_{q} + r)^{c_{j}} = (m[0]_{q} + r)^{c_{0}}(m[1]_{q} + r)^{c_{1}}(m[2]_{q} + r)^{c_{2}} ⋯ (m[k]_{q} + r)^{c_{k}}  (29)

in the explicit formula in (25), we observe that there are exactly n − k factors of (m[j]_{q} + r) which is repeated c_{j} times for each j. From here, we write

(m[0]_{q} + r)^{c_{0}} = (m[j_{1}]_{q} + r)(m[j_{2}]_{q} + r) ⋯ (m[j_{c_{0}}]_{q} + r),

where j_{i} = 0, i = 1, 2, …, c_{0};

(m[1]_{q} + r)^{c_{1}} = (m[j_{c_{0}+1}]_{q} + r)(m[j_{c_{0}+2}]_{q} + r) ⋯ (m[j_{c_{0}+c_{1}}]_{q} + r),

where j_{c_{0}+i} = 1, i = 1, 2, …, c_{1};

(m[2]_{q} + r)^{c_{2}} = (m[j_{c_{0}+c_{1}+1}]_{q} + r)(m[j_{c_{0}++c_{1}+2}]_{q} + r) ⋯ (m[j_{c_{0}+c_{1}+c_{2}}]_{q} + r),

where j_{c_{0}+c_{1}+i} = 2, i = 1, 2, …, c_{2} and so on until

(m[k]_{q} + r)^{c_{k}} = (m[j_{c_{0}+c_{1}+···+c_{k−1}+1}]_{q} + r)(m[j_{c_{0}+c_{1}+···+c_{k−1}+2}]_{q} + r) ⋯ (m[j_{c_{0}+c_{1}+···+c_{k}}]_{q} + r),

where j_{c_{0}+c_{1}+···+c_{k−1}+i} = k, i = 1, 2, …, c_{k} and c_{0} + c_{1} + c_{2} + ⋯ + c_{k−1} + c_{k} = n − k. Thus, 0 ≤ j_{1} ≤ j_{2} ≤ ⋯ ≤ j_{n−k} ≤ k and we have

W_{m,r,q}(n, k) = q^{\binom{k}{2}} ∑_{0≤j_{1}≤j_{2}≤···≤j_{n−k}≤k} ∏_{i=1}^{n−k}(m[j_{i}]_{q} + r).  (30)

We formally state this result in the next theorem.

Theorem 4. The (q, r)-Whitney numbers of the second kind satisfy the following explicit form:

W_{m,r,q}(n, k) = q^{\binom{k}{2}} ∑_{0≤j_{1}≤j_{2}≤···≤j_{n−k}≤k} ∏_{i=1}^{n−k}(m[j_{i}]_{q} + r).  (31)

Notice that when q → 1, we obtain an identity similar to the result obtained by Mangontarum et al. [18, Theorem 11].

3 On the context of A-tableaux

De Medicis and Leroux [23] defined a 0-1 tableau to be a pair φ = (λ, f), where λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{k}) is a partition of an integer m and f = (f_{ij})_{1≤j≤λ_{i}} is a “filling” of the cells of the corresponding Ferrers diagram of shape λ with 0’s and 1’s such that exactly one 1 in each

0 0 0 1 0 1 0 1
0 1 0 0 1 0 1
0 0 1 0 0
1 0 0 0

0

Figure 1: A 0-1 tableau φ

column. For instance, Figure 1 represents the 0-1 tableau φ = (λ, f), where λ = (8, 7, 5, 4, 1) with

f_{14} = f_{16} = f_{18} = f_{22} = f_{25} = f_{27} = f_{33} = f_{41} = 1

and f_{ij} = 0 elsewhere for 1 ≤ j ≤ λ_{i}. In the same paper, an A-tableau is defined to be a list Φ of columns c of a Ferrers diagram of a partition λ (by decreasing order of length) such that the length |c| is part of the sequence A = (a_{i})_{i≥0}, a strictly increasing sequences of non-negative integers. Combinatorial interpretations of Stirling-type numbers in terms of A-tableaux are already done in the past. Similar works can be seen in [9, 12, 14, 17, 23] and some of the references therein. In particular, Corcino and Montero [14] defined a q-analogue of the Rucinski-Voigt numbers (an equivalent of the r-Whitney numbers of the second kind) and then presented a combinatorial interpretation using the theory of A-tableaux. The same type of interpretation was obtained by Mangontarum et al. [17] for the case of the translated Whitney numbers (see [20]) and their q-analogues. It is important to note that the q-analogues of these authors follow motivations which differ from that of the (q, r)-Whitney numbers. Furthermore, the numbers considered in the paper of Ramírez and Shattuck [26] belong to p, q-analogues, a natural extension of q-analogues.

Now, we let ω be a function from the set of non-negative integers N to a ring K, and suppose that Φ is an A-tableau with r columns of length |c|. Also, it is known that Φ might contain a finite number of columns whose lengths are zero since 0 ∈ A and if ω(0) ≠ 0 (cf. [23]). Before proceeding, we denote by T^{A}(x, y) the set of A-tableaux with A = {0, 1, 2, …, x} and exactly y columns (with some columns possibly of zero length), and by T_{d}^{A}(x, y) the subset of T^{A}(x, y) which contains all A-tableaux with columns of distinct lengths. The next theorem relates the (q, r)-Whitney numbers of both kinds to certain sets of A-tableaux.

Theorem 5. Let Ω : N ⟶ K and ω : N ⟶ K be functions from the set of non-negative integers N to a ring K (column weights according to length) defined by

Ω(|c|) = m[|c|]_{q} + r

and

ω(|c|) = m[|c̄|]_{q} + r,

where m and r are complex numbers, |c| is the length of column c of an A-tableau in T_{d}^{A}(n − 1, n − k), and |c̄| is the length of column c of an A-tableau in T^{A}(k, n − k). Then

(−1)^{n−k}q^{\binom{n}{2}}w_{m,r,q}(n, k) = ∑_{Φ∈T_{d}^{A}(n−1,n−k)} ∏_{c∈Φ} Ω(|c|)  (32)

and

q^{−\binom{k}{2}}W_{m,r,q}(n, k) = ∑_{ϕ∈T^{A}(k,n−k)} ∏_{c̄∈ϕ} ω(|c̄|).  (33)

Proof. Let Φ ∈ T_{d}^{A}(n − 1, n − k). This means that Φ has exactly n − k columns, say c_{1}, c_{2}, …, c_{n−k} whose lengths are j_{1}, j_{2}, …, j_{n−k}, respectively. Now, for each column c_{i} ∈ Φ, i = 1, 2, 3, …, n − k, we have |c_{i}| = j_{i} and

Ω(|c_{i}|) = m[|j_{i}|]_{q} + r.

Thus,

∏_{c∈Φ} Ω(|c|) = ∏_{i=1}^{n−k} Ω(|c_{i}|)

= ∏_{i=1}^{n−k}(m[j_{i}]_{q} + r).

Since Φ ∈ T_{d}^{A}(n − 1, n − k), then

∑_{Φ∈T_{d}^{A}(n−1,n−k)} ∏_{c∈Φ} Ω(|c|) = ∑_{0≤j_{1}<j_{2}<···<j_{n−k}≤n−1} ∏_{c∈Φ} Ω(|c|)

= ∑_{0≤j_{1}<j_{2}<···<j_{n−k}≤n−1} ∏_{i=1}^{n−k}(m[j_{i}]_{q} + r)
= (−1)^{n−k}q^{\binom{n}{2}}w_{m,r,q}(n, k).

The second result is obtained similarly.

3.1 Combinatorics of A-tableaux

In the following theorem, we will demonstrate the simple combinatorics of A-tableaux. To start, note that Eqs. (32) and (33) are equivalent to

(−1)^{n−k}q^{\binom{n}{2}}w_{m,r,q}(n, k) = ∑_{Φ∈T_{d}^{A}(n−1,n−k)} Ω_{A}(Φ)  (34)

and

q^{−\binom{k}{2}}W_{m,r,q}(n, k) = ∑_{ϕ∈T^{A}(k,n−k)} ω_{A}(ϕ),  (35)

respectively, where

Ω_{A}(Φ) = ∏_{c∈Φ} Ω(|c|) = ∏_{c∈Φ}(m[|c|]_{q} + r), |c| ∈ {0, 1, 2, …, n − 1}  (36)

and

ω_{A}(ϕ) = ∏_{c̄∈ϕ} ω(|c̄|) = ∏_{c̄∈ϕ}(m[|c|]_{q} + r), |c̄| ∈ {0, 1, 2, …, k}.  (37)

Theorem 6. For nonnegative integers n and k, and complex numbers m and r, the following identities hold:

w_{m,r,q}(n, k) = ∑_{j=k}^{n} \binom{j}{k}(−r_{2})^{j−k}w_{m,r_{1},q}(n, j)  (38)

W_{m,r,q}(n, k) = ∑_{j=k}^{n} \binom{n}{j}r_{2}^{n−j}W_{m,r_{1},q}(j, k),  (39)

where r_{1} + r_{2} = r.

Proof. Let Φ ∈ T_{d}^{A}(n − 1). Substituting j_{i} = |c| in Eq. (36) gives

Ω_{A}(Φ) = ∏_{i=1}^{n−k}(m[j_{i}]_{q} + r),

where j_{i} ∈ {0, 1, 2, …, n − 1}. Suppose that for some numbers r_{1} and r_{2}, r = r_{1} + r_{2}. Then, with Ω^{∗}(j_{i}) = m[j_{i}]_{q} + r_{1}, we may write

Ω_{A}(Φ) = ∏_{i=1}^{n−k} [(m[j_{i}]_{q} + r_{1}) + r_{2}]

= ∏_{i=1}^{n−k}(Ω^{∗}(j_{i}) + r_{2})

= ∑_{ℓ=0}^{n−k} r_{2}^{n−k−ℓ} ∑_{j_{1}≤q_{1}<q_{2}<···<q_{ℓ}≤j_{n−k}} ∏_{i=1}^{ℓ} Ω^{∗}(q_{i}).

Let B_{Φ} be the set of all A-tableaux corresponding to Φ such that for each ψ ∈ B_{Φ}, one of the following is true:

ψ has no column whose weight is r_{2}; ψ has one column whose weight is r_{2};

ψ has two columns whose weight are r_{2};

⋮

ψ has n − k columns whose weight are r_{2}.

Then,

Ω_{A}(Φ) = ∑_{ψ∈B_{Φ}} Ω_{A}(ψ).

Now, if ℓ columns in ψ have weights other than r_{2}, then

Ω_{A}(ψ) = ∏_{c∈ψ} Ω^{∗}(|c|)

= r_{2}^{n−kℓ} ∏_{i=1}^{ℓ} Ω^{∗}(q_{i}),

where q_{1}, q_{2}, q_{3}, …, q_{ℓ} ∈ {j_{1}, j_{2}, j_{3}, …, j_{n−k}}. Hence, Eq. (34) may be written as

(−1)^{n−k}q^{\binom{n}{2}}w_{m,r,q}(n, k) = ∑_{Φ∈T_{d}^{A}(n−1,n−k)} Ω_{A}(Φ)
= ∑_{Φ∈T_{d}^{A}(n−1,n−k)} ∑_{ψ∈B_{Φ}} Ω_{A}(ψ).

For each ℓ, it is known that there correspond \binom{n−k}{ℓ} tableaux with ℓ distinct columns with weights Ω^{∗}(q_{i}), q_{i} ∈ \{j_{1}, j_{2}, …, j_{n−k}\}. Since T_{d}^{A}(n − 1, n − k) contains \binom{n}{k} tableaux, then for each Φ ∈ T_{d}^{A}(n − 1, n − k), the total number of A-tableaux ψ corresponding to Φ is

\binom{n}{k}\binom{n − k}{ℓ}.

However, only \binom{n}{ℓ} tableaux in B_{Φ} with ℓ distinct columns of weights other than r_{2} are distinct. It then follows that every distinct tableau ψ appears

\begin{matrix}\binom{n}{k} & \binom{n−k}{ℓ} \\ ( & {n \atop ℓ} & ) \end{matrix} = \binom{n − ℓ}{k}

times in the collection (cf. [12]). Thus, we consequently obtain

(−1)^{n−k}q^{\binom{n}{2}}w_{m,r,q}(n, k) = ∑_{ℓ=0}^{n−k} \binom{n − ℓ}{k}r_{2}^{n−k−ℓ} ∑_{ψ∈B_{ℓ}} ∏_{c∈ψ} Ω^{∗}(|c|),

where B_{ℓ} denotes the set of all tableaux ψ having ℓ distinct columns whose lengths are in the set {0, 1, 2, …, n − 1}. Reindexing the double sum yields

(−1)^{n−k}q^{\binom{n}{2}}w_{m,r,q}(n, k) = ∑_{j=k}^{n} \binom{j}{k}r_{2}^{j−k} ∑_{ψ∈B_{n−j}} ∏_{c∈ψ} Ω^{∗}(|c|).  (40)

Since B_{n−j} = T_{d}^{A}(n − 1, n − j), then

∑_{ψ∈B_{n−j}} ∏_{c∈ψ} Ω^{∗}(|c|) = (−1)^{n−j}q^{\binom{n}{2}}w_{m,r_{1},q}(n, j).  (41)

Combining Eqs. (40) and (41) gives

(−1)^{n−k}q^{\binom{n}{2}}w_{m,r,q}(n, k) = ∑_{j=k}^{n} \binom{j}{k}r_{2}^{j−k}(−1)^{n−j}q^{\binom{n}{2}}w_{m,r_{1},q}(n, j)  (42)

which is equivalent to the desired result in Eq. (38). Similarly, if ϕ ∈ T^{A}(n − 1), then substituting j_{i} = |c̄| in Eq. (37) gives

ω_{A}(ϕ) = ∏_{i=1}^{n−k}(m[j_{i}]_{q} + r),

where j_{i} ∈ {0, 1, 2, …, k}. If for some numbers r_{1} and r_{2}, r = r_{1} + r_{2}, then

ω_{A}(ϕ) = ∏_{i=1}^{n−k} [(m[j_{i}]_{q} + r_{1}) + r_{2}]

= ∏_{i=1}^{n−k}(ω^{∗}(j_{i}) + r_{2}), ω^{∗}(j_{i}) = m[j_{i}]_{q} + r_{1}

= ∑_{ℓ=0}^{n−k} r_{2}^{n−k−ℓ} ∑_{j_{1}≤q_{1}≤q_{2}≤···≤q_{ℓ}≤j_{n−k}} ∏_{i=1}^{ℓ} ω^{∗}(q_{i}).

Suppose B̄_{ϕ} is the set of all A-tableaux corresponding to ϕ such that for each ζ ∈ B̄_{ϕ}, one of the following is true:

ζ has no column whose weight is r_{2};

ζ has one column whose weight is r_{2};

ζ has two columns whose weight are r_{2};

⋮ ζ has n − k columns whose weight are r_{2}.

Then, we may write

ω_{A}(ϕ) = ∑_{ζ∈B̄_{ϕ}} ω_{A}(ζ).

If there are ℓ columns in ζ with weights other than r_{2}, then we have

ω_{A}(ζ) = ∏_{c̄∈ζ} ω^{∗}(|c̄|)

= r_{2}^{n−kℓ} ∏_{i=1}^{ℓ} ω^{∗}(q_{i}),

where q_{1}, q_{2}, q_{3}, …, q_{ℓ} ∈ {j_{1}, j_{2}, j_{3}, …, j_{n−k}}. It then follows that Eq. (35) may be expressed as

q^{−\binom{k}{2}}W_{m,r,q}(n, k) = ∑_{ϕ∈T^{A}(k,n−k)} ω_{A}(ϕ)
= ∑_{ϕ∈T^{A}(k,n−k)} ∑_{ζ∈B̄_{ϕ}} ω_{A}(ζ).

Like in the previous, for each ℓ, there correspond \binom{n−k}{ℓ} tableaux with ℓ columns having weights ω^{∗}(q_{i}), q_{i} ∈ \{j_{1}, j_{2}, …, j_{n−k}\}. Since the set T^{A}(k, n − k) contains \binom{n}{k} tableaux, then for each ϕ ∈ T^{A}(k, n − k), there are

\binom{n}{k}\binom{n − k}{ℓ}

A-tableaux corresponding to ϕ. But only \binom{ℓ+k}{ℓ} of these tableaux are distinct. Hence, every distinct tableau ζ with ℓ columns of weights other than r_{2} appears

\begin{matrix}\binom{n}{k} & \binom{n−k}{ℓ} \\ ( & {ℓ+k \atop ℓ} & ) \end{matrix} = \binom{n}{ℓ + k}

times in the collection (cf. [9]). It implies that

q^{−\binom{k}{2}}W_{m,r,q}(n, k) = ∑_{ℓ=0}^{n−k} \binom{n}{ℓ + k}r_{2}^{n−k−ℓ} ∑_{ζ∈B̄_{ℓ}} ∏_{c̄∈ζ} ω^{∗}(|c̄|),

where B̄_{ℓ} is the set of all tableaux ζ having ℓ columns of weights ω^{∗}(j_{i}). Reindexing the sums yield

q^{−\binom{k}{2}}W_{m,r,q}(n, k) = ∑_{j=k}^{n} \binom{n}{j}r_{2}^{n−j} ∑_{ζ∈B̄_{j−k}} ∏_{c̄∈ζ} ω^{∗}(|c̄|).  (43)

Since B̄_{n−j} = T^{A}(k, n − j), then

∑_{ζ∈B̄_{j−k}} ∏_{c̄∈ζ} ω^{∗}(|c̄|) = q^{−\binom{k}{2}}W_{m,r_{1},q}(j, k).  (44)

Moreover, by Eqs. (43) and (44), we obtain

q^{−\binom{k}{2}}W_{m,r,q}(n, k) = ∑_{j=k}^{n} \binom{n}{j}r_{2}^{n−j}q^{−\binom{k}{2}}W_{m,r_{1},q}(j, k)  (45)

which is equivalent to the second desired result.

Let r_{1} = r − 1 and r_{2} = 1 in Eqs. (38) and (39). Then.

w_{m,r,q}(n, k) = ∑_{j=k}^{n} \binom{j}{k}(−1)^{j−k}w_{m,r−1,q}(n, j)  (46)

and

W_{m,r,q}(n, k) = ∑_{j=k}^{n} \binom{n}{j}W_{m,r−1,q}(j, k).  (47)

These identities were first seen in [21, Theorem 9]. Now, using Eq. (39), the (q, r)-Dowling numbers D_{m,r,q}(n) [21] may be expressed as

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

Moreover, by applying the binomial inversion formula [8]

f_{n} = ∑_{j=0}^{n} \binom{n}{j}g_{j} ⟺ g_{n} = ∑_{j=0}^{n}(−1)^{n−j}\binom{n}{j}f_{j}

to this identity gives

D_{m,r−1,q}(n) = ∑_{j=0}^{n}(−1)^{n−j}\binom{n}{j}D_{m,r,q}(j).

These results are formally stated in the following corollary:

Corollary 7. The (q, r)-Dowling numbers satisfy the recurrence relations with respect to r given by

D_{m,r+1,q}(n) = ∑_{j=0}^{n} \binom{n}{j}D_{m,r,q}(j)  (48)

and

D_{m,r,q}(n) = ∑_{j=0}^{n}(−1)^{n−j}\binom{n}{j}D_{m,r+1,q}(j).  (49)

Remark 8. When q → 1 and m = β, we obtain the following identities by Corcino and Corcino [11]:

G_{n,β,r+1} = ∑_{j=0}^{n} \binom{n}{j}G_{j,β,r}  (50)

G_{n,β,r} = ∑_{j=0}^{n}(−1)^{n−j}\binom{n}{j}G_{j,β,r+1},  (51)

where G_{n,β,r} := D_{β,r,1}(n) is the generalized Bell numbers in [10, 11]. These identities were used to identify the Hankel transform of G_{n,β,r}.

Looking at the previous corollary, we see that the sequence (D_{m,r+1,q}(n)) is the binomial transform of the sequence (D_{m,r,q}(n)), for r = 0, 1, 2, . . .. Using “Layman’s Theorem” [16], (D_{m,0,q}(n)), (D_{m,1,q}(n)), (D_{m,2,q}(n)), …, (D_{m,r,q}(n)), … have the same Hankel transform. This directs our attention to the following open problem:

Problem 9. Is it possible to identify the Hankel transform of D_{m,r,q}(n) using a method parallel to what is being done in [11] for G_{n,β,r}?

3.2 Convolution-type identities

Recall that for any two sequences a_{n} and b_{n}, we call the sequence c_{n} as convolution sequence if

c_{n} = ∑_{k=1}^{n} a_{n}b_{n−k}, n = 0, 1, 2, ….  (52)

One of the most famous convolution-type identity is the Vandermonde’s formula [6, 8] given by

\binom{a + b}{n} = ∑_{k=0}^{n} \binom{a}{k}\binom{b}{n − k}.  (53)

The following theorem contains convolution-type identities for the (q, r)-Whitney numbers of the first kind which will be proved using the combinatorics of A-tableaux:

Theorem 10. The (q, r)-Whitney numbers of the first kind have convolution-type identities given by

w_{m,r,q}(p + j, n) = q^{−pj} ∑_{k=0}^{n} w_{m,r,q}(p, k)w_{m̄,r̄,q}(j, n − k)  (54)

and

w_{m,r,q}(n + 1, j + p + 1) = ∑_{k=0}^{n} q^{k^{2}−nk−n}w_{m,r,q}(k, p)w_{m̄,r̄,q}(n − k, j),  (55)

where m̄ = mq^{p} and r̄ = m[p]_{q} + r.

Proof. For A_{1} = {0, 1, 2, …, p − 1} and A_{2} = {p, p + 1, p + 2, …, p + j − 1}, let Φ_{1} ∈ T_{d}^{A_{1}}(p − 1, p − k) and Φ_{2} ∈ T_{d}^{A_{2}}(j − 1, j − n + k). Note that by joining the columns of the tableaux Φ_{1} and Φ_{2}, we may generate an A-tableau Φ with p + j − n distinct columns whose lengths are in the set A = {0, 1, 2, …, p + j − 1}. That is, Φ ∈ T_{d}^{A}(p + j − 1, p + j − n). Hence,

∑_{Φ∈T_{d}^{A}(p+j−1,p+j−n)} Ω_{A}(Φ) = ∑_{k=0}^{n} {  ∑_{Φ_{1}∈T_{d}^{A_{1}}(p−1,p−k)} Ω_{A_{1}}(Φ_{1})} {  ∑_{Φ_{2}∈T_{d}^{A_{2}}(j−1,j−n+k)} Ω_{A_{2}}(Φ_{2})} .

Note that in the right-hand side, we get

∑_{Φ_{2}∈T_{d}^{A_{2}}(j−1,j−n+k)} Ω_{A_{2}}(Φ_{2}) = ∑_{p≤g_{1}<g_{2}<···<g_{j−n+k}≤p+j−1} ∏_{i=1}^{j−n+k}(m[g_{i}]_{q} + r)

= ∑_{0≤g_{1}<g_{2}<···<g_{j−n+k}≤j−1} ∏_{i=1}^{j−n+k}(m[p + g_{i}]_{q} + r)

= ∑_{0≤g_{1}<g_{2}<···<g_{j−n+k}≤j−1} ∏_{i=1}^{j−n+k} (mq^{p}[g_{i}]_{q} + ([p]_{q} + r))
= (−1)^{j−n+k}q^{\binom{j}{2}}w_{m̄,r̄,q}(j, n − k),

where m̄ = mq^{p} and r̄ = m[p]_{q} + r. Also, using Eq. (34),

∑_{Φ_{1}∈T_{d}^{A_{1}}(p−1,p−k)} Ω_{A_{1}}(Φ_{1}) = (−1)^{p−k}q^{\binom{p}{2}}w_{m,r,q}(p, k)

and

∑_{Φ∈T_{d}^{A}(p+j−1,p+j−n)} Ω_{A}(Φ) = (−1)^{p+j−n}q^{\binom{p+j}{2}}w_{m,r,q}(p + j, n).

Hence, by simplification, we obtain the convolution identity (54). Similarly, we let Φ_{1} be a tableau with k − p columns whose lengths are in B_{1} = {0, 1, 2, …, k − 1} and Φ_{2} be a tableau with n − k − j columns whose lengths are in B_{2} = {k + 1, k + 2, …, n} so that Φ ∈ T_{d}^{B_{1}}(k − 1, k − p) and Φ ∈ T_{d}^{B_{2}}(n − k − 1, n − k − j). Note that we may generate an A-tableau Φ by joining the columns of Φ_{1} and Φ_{2} whose lengths are in A = {0, 1, 2, …, n}. Hence, we have

∑_{Φ∈T_{d}^{A}(n,n−j−p)} Ω_{A}(Φ) = ∑_{k=0}^{n} {  ∑_{Φ_{1}∈T_{d}^{B_{1}}(k−1,k−p)} Ω_{B_{1}}(Φ_{1})} {  ∑_{Φ_{2}∈T_{d}^{B_{2}}(n−k−11,n−k−j)} Ω_{B_{2}}(Φ_{2})} .

Applying Eq. (34) gives

∑_{Φ∈T_{d}^{A}(n,n−j−p)} Ω_{A}(Φ) = (−1)^{n−j−p}q^{\binom{n+1}{2}}w_{m,r,q}(n + 1, j + p + 1)

and

∑_{Φ_{1}∈T_{d}^{B_{1}}(k−1,k−p)} Ω_{B_{1}}(Φ_{1}) = (−1)^{k−p}q^{\binom{k}{2}}w_{m,r,q}(k, p).

Also, in the right-hand side, we get

∑_{Φ_{2}∈T_{d}^{B_{2}}(n−k−11,n−k−j)} Ω_{B_{2}}(Φ_{2}) = ∑_{p≤g_{1}<g_{2}<···<g_{n−k−j}≤p+n−k−1} ∏_{i=1}^{n−k−j}(m[g_{i}]_{q} + r)

= ∑_{0≤g_{1}<g_{2}<···<g_{n−k−j}≤n−k−1} ∏_{i=1}^{n−k−j}(m[p + g_{i}]_{q} + r)

= ∑_{0≤g_{1}<g_{2}<···<g_{n−k−j}≤n−k−1} ∏_{i=1}^{n−k−j} (mq^{p}[g_{i}]_{q} + ([p]_{q} + r))
= (−1)^{n−k−j}q^{\binom{n−k}{2}}w_{m̄,r̄,q}(n − k, j),

where m̄ = mq^{p} and r̄ = m[p]_{q} + r. This completes the proof.

The next theorem can be proved similarly.

Theorem 11. The (q, r)-Whitney numbers of the second kind have convolution-type identities given by

W_{m,r,q}(n + 1, j + p + 1) = ∑_{k=0}^{n} q^{p+pj+j}W_{m,r,q}(k, p)W_{m̂,r̂,q}(n − k, j)  (56)

and

W_{m,r,q}(p + j, n) = ∑_{k=0}^{n} q^{nk−k^{2}}W_{m,r,q}(p, k)W_{m̂,r̂,q}(j, n − k),  (57)

where m̂ = mq^{p+1} and r̂ = m[p + 1]_{q} + r.

As q → 1, we recover from Theorems 10 and 11 the results recently obtained by Xu and Zhou [27, Theorems 2.1 and 2.4].

4 On Heine and Euler distributions

Consider the Poisson distribution

f_{X}(x) = e^{−λ} \frac{λ^{x}}{x!} ,  (58)

for x = 0, 1, 2, . . .. The factorial moment of a Poisson random variable is readily evaluated, i.e.,

E[(X)_{n}] = λ^{n}  (59)

the mean, E[X] = λ, being the special case n = 1. Expanding x^{n} in terms of falling factorials (using the Stirling numbers of the second kind), we obtain the n-th moment of X given by

E[X^{n}] = B_{n}(λ),  (60)

where B_{n}(λ) are the Bell polynomials. The q-analogues of the Poisson distribution introduced by Kemp [15], and Benkherouf and Bather in [3] are given by

f_{Y} (y) = e_{q}(−λ)q^{\binom{y}{2}} \frac{λ^{y}}{[y]_{q}!}, y = 0, 1, 2, …  (61)

and

f_{Z}(z) = ê_{q}(−λ)\frac{λ^{z}}{[z]_{q}!}, z = 0, 1, 2, ….  (62)

These are called Heine and Euler distributions, respectively, where

e_{q}(t) = ∑_{k=0}^{∞} \frac{t^{k}}{[k]_{q}!}  (63)

and

ê_{q}(t) = ∑_{k=0}^{∞} q^{\binom{k}{2}} \frac{t^{k}}{[k]_{q}!}.  (64)

In line with this, Charalambides and Papadatos [5] obtained the following important results:

E[[Y]_{r,q}] = \frac{q^{\binom{r}{2}}λ^{r}}{∏_{i=1}^{r}(1 + λ(1 − q)q^{i−1})},  (65)

E[[Z]_{r,q}] = λ^{r},  (66)

where [x]_{r,q} = [x]_{q}[x − 1]_{q}[x − 2]_{q} ⋯ [x − r + 1]_{q} is the q-falling factorial of x of order r. Considering these, we now state the following theorem:

Theorem 12. If Y and Z are random variables with Heine and Euler distributions, respectively, and if the mean of Y is ϕ = \frac{λ}{1+λ(1−q)} and the mean of Z is λ, then

E_{ϕ}[(m[Y]_{q} + r)^{n}] = ∑_{ℓ=0}^{n} ∑_{i=0}^{n}(−λ)^{i}q^{−\binom{ℓ}{2}−ℓi} \frac{λ^{ℓ}}{[ℓ]_{q}![i]_{q}!} \frac{(m[ℓ]_{q} + r)^{n}}{∏_{j=1}^{ℓ+i}(1 + λ(1 − q)q^{j−1})},  (67)

E_{λ}[(m[Z]_{q} + r)^{n}] = ê_{q}(−λ) ∑_{ℓ=0}^{n} \frac{λ^{ℓ}}{[ℓ]_{q}!}(m[ℓ]_{q} + r)^{n}.  (68)

Proof. From the defining relation in (1) and the result in (65),

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

Using the explicit formula for the (q, r)-Whitney numbers of the second kind [21, Theorem 16] given by

W_{m,r,q}(n, k) = \frac{1}{m^{k}[k]_{q}!} ∑_{ℓ=0}^{k}(−1)^{k−ℓ}q^{\binom{k−ℓ}{2}}\binom{k}{ℓ}_{q}(m[ℓ]_{q} + r)^{n},  (69)

we obtain

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

×\frac{q^{\binom{k}{2}}λ^{k}}{∏_{j=1}^{k}(1 + λ(1 − q)q^{j−1})}

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

Reindexing the second sum yields (67). Eq. (68) may be shown similarly.

Remark 13. When m = 1 and r = 0 in the previous theorem, we have

E_{ϕ}[[Y]_{q}^{n}] = ∑_{ℓ=0}^{n} ∑_{i=0}^{n}(−λ)^{i}q^{−\binom{ℓ}{2}−ℓi} \frac{λ^{ℓ}}{[ℓ]_{q}![i]_{q}!} \frac{[ℓ]_{q}^{n}}{∏_{j=1}^{ℓ+i}(1 + λ(1 − q)q^{j−1})},  (70)

and

E_{λ}[[Z]_{q}^{n}] = ê_{q}(−λ) ∑_{ℓ=0}^{n} \frac{λ^{ℓ}}{[ℓ]_{q}!}[ℓ]_{q}^{n} ≡ B_{n,q}(λ),  (71)

where B_{n,q}(λ) is the q-Bell polynomials. On the other hand, if the mean is λ = \frac{x}{m},

E_{x/m} \{(m[Z]_{q} + r)^{n}\} = ê_{q} (−\frac{x}{m}) ∑_{ℓ=0}^{n} \frac{x^{ℓ}}{m^{ℓ}} \frac{(m[ℓ]_{q} + r)^{n}}{[ℓ]_{q}!} .

This explicit formula is due to Mangontarum and Katriel [21]. Thus

E_{x/m}[(m[Z]_{q} + r)^{n}] = D_{m,r,q}(n, x),

where

D_{m,r,q}(n, x) = ∑_{k=0}^{n} W_{m,r,q}(n, k)x^{k}  (72)

is the (q, r)-Dowling polynomials.

It is worth mentioning that Mangontarum and Corcino [19] obtained the following pair of n-th order generalized factorial moments

E_{λ}[(βX + γ|α)_{n}] = e^{−λ} ∑_{i=0}^{∞} \frac{(iβ + γ|α)_{n}}{i!} λ^{i}  (73)

E_{λ}[(αX − γ|β)_{n}] = e^{−λ} ∑_{i=0}^{∞} \frac{(iα − γ|β)_{n}}{i!} λ^{i},  (74)

where X is a Poisson random variable with mean λ and α, β and γ may be real or complex numbers. Here,

(t|α)_{n} = t(t − α)(t − 2α) ⋯ (t − nα + α),  (75)

with initial conditions (t| α)_{n} = 0 when n ≤ 0 and (t| α)_{0} = 1. Notice that (73) unifies the factorial moment in (59) and the n-th moment in (60). More precisely,

• when β = 1, γ = 0 and α = 0,

E_{λ}[(βX + γ|α)_{n}] = E_{λ}[X^{n}];

• when β = 1, γ = 0 and α = 1,

E_{λ}[(βX + γ|α)_{n}] = E_{λ}[(X)_{n}].

Other known “Bell-type” and “Dowling-type” polynomials (see [7, 10, 18, 22, 24, 25]) can be shown to be particular cases of Eqs. (73) and (74). Furthermore, Corcino and Mangontarum [13] obtained the generalized q-factorial moments

E_{ϕ}[ [[βY]_{q} + [γ]_{q}|[α]_{q}]_{n,q} ] = ∑_{j=0}^{∞} ê_{q^{β},j}(−λ)\frac{(q^{β}λ)^{j} [[βj]_{q} + [γ]_{q}|[α]_{q}]_{n,q}}{[j]_{q^{β}}! ∏_{i=1}^{j} (1 + λ(1 − q^{β})q^{β(i−1)})}  (76)

and

E_{λ}[ [[βZ]_{q} + [γ]_{q}|[α]_{q}]_{n,q} ] = ê_{b}(−λ) ∑_{j=0}^{∞} [[βj]_{q} + [γ]_{q}|[α]_{q}]_{n,q} \frac{λ^{j}}{[j]_{b}!},  (77)

where Y is a random variable with Heine distribution and mean ϕ = \frac{λ}{1+λ(1−q^{β})}, and Z is a random variable with an Euler distribution and mean λ. The notations

[[βZ]_{q} + [γ]_{q}|[α]_{q}]_{n,q} = ∏_{j=0}^{n−1} ([βt]_{q} + [γ]_{q} − [αj]_{q})  (78)

and

ê_{q^{β},j}(−λ) = ∑_{l=0}^{∞} [\frac{q^{β\binom{j}{2}}(−λ)^{l}}{[l]_{q^{β}}! ∏_{i=1}^{l} (q^{β(i−1)} + λ(1 − q^{β})q^{βj})}]  (79)

are used. (76) and (77) are found to be q-analogues of (73). By thoroughly investigating (68), it is obvious that this result is not generalized by (76) and (77).

Privault [25] defined an extension of the classical Bell numbers as

e^{ty−λ(e^{t}−t−1)} = ∑_{k=0}^{∞} B_{n}(y, λ)\frac{t^{k}}{k!}.

Moreover, he obtained the following n-th moment of a Poisson random variable

E_{λ}[(X + y − λ)^{n}] = B_{n}(y, −λ),  (80)

where

B_{n}(y, −λ) = ∑_{k=0}^{n} \binom{n}{k}(y − λ)^{n−k} ∑_{j=0}^{k} \left\{{k \atop j}\right\}λ^{j},  (81)

Corcino and Corcino [10] showed that the (r, β)-Bell polynomials satisfy

G_{n,β,r}(x) = ∑_{k=0}^{n} \binom{n}{k}r^{n−k} ∑_{j=0}^{k} β^{k−j}\left\{{k \atop j}\right\}x^{j}.  (82)

It then follows that

G_{n,1,y−λ}(λ) = B_{n}(y, −λ).

The next theorem is analogous to these identities.

Theorem 14. The (q, r)-Dowling polynomials satisfy the identity

D_{m,r,q}(n, x) = ∑_{k=0}^{n} \binom{n}{k}r^{n−k} ∑_{j=0}^{k} m^{k−j}\left\{{k \atop j}\right\}_{q}x^{j}.  (83)

Proof. Using the binomial theorem, we have

E_{x/m}[(m[Z]_{q} + r)^{n}] = ∑_{k=0}^{n} \binom{n}{k}r^{n−k}m^{k}E_{x/m}[[Z]_{q}^{k}]

= ∑_{k=0}^{n} \binom{n}{k}r^{n−k}m^{k}B_{n,q} (\frac{x}{m})

= ∑_{k=0}^{n} \binom{n}{k}r^{n−k}m^{k} ∑_{j=0}^{k} \left\{{k \atop j}\right\}_{q} (\frac{x}{m})^{j} .

The desired result follows from the fact that E_{x/m}[(m[Z]_{q} + r)^{n}] = D_{m,r,q}(n, x).

Remark 15. As q → 1, we obtain the (r, β)-Bell polynomial identity in Eq. (82). If the mean is replaced with λ, then for an Euler random variable Z,

E_{λ}[(m[Z]_{q} + r)^{n}] = ∑_{k=0}^{n} \binom{n}{k}r^{n−k} ∑_{j=0}^{k} m^{k}\left\{{k \atop j}\right\}_{q}λ^{j}.

As q → 1, we get [19, Eq. 34]

E_{λ}[(mX + r)^{n}] = ∑_{k=0}^{n} \binom{n}{k}r^{n−k} ∑_{j=0}^{k} m^{k}\left\{{k \atop j}\right\}λ^{j}.

When m = 1 and r = y − λ,

D_{1,y−λ,q}(n, x) = ∑_{k=0}^{n} \binom{n}{k}(y − λ)^{n−k} ∑_{j=0}^{k} \left\{{k \atop j}\right\}_{q}x^{j}.  (84)

This is a q-analogue of Privault’s identity since (84) → (81) as q → 1.

5 Acknowledgment

The author is thankful to the editor-in-chief and to the referee(s) for reading carefully the paper and giving helpful comments and suggestions. Special thanks also to Dr. Jacob Katriel for sharing his insights during the early stage of this paper. This research is supported by the Mindanao State University through the Special Order No. 26 — OP, Series of 2016.

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2010 Mathematics Subject Classification: Primary 11B83; Secondary 11B73, 05A30. Keywords: (q, r)-Whitney number, symmetric function, A-tableau, q-distribution.

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

Received August 16 2016; revised versions received November 20 2016; November 21 2016. Published in Journal of Integer Sequences, December 27 2016.

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