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Journal of Integer Sequences, Vol. 18 (2015),
Article 15.9.8
23 11
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
1
number, the q-factorial and the q-falling factorial of order r, respectively, given by
n
r−1
Y
Y
qx − 1
, [n]q ! =
[x]q =
[i]q , [x]q,n =
[x − i]q ,
q−1
i=1
i=0
for any real number x and non-negative integers n and r, and the q-binomial coefficients
(also known as Gaussian polynomials)
[n]q !
n
[n]q,r
=
=
.
r q [r]q ![n − r]q !
[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 Qn (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 Qn (G),
denoted by wm (n, k) and Wm (n, k), respectively, in terms of the relations
n
m (x)n =
n
X
wm (n, k)(mx + 1)k
(1)
k=0
and
n
(mx + 1) =
n
X
mk Wm (n, k)(x)k ,
(2)
k=0
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
nfor the well-known Stirling numbers of the
n
first and second kind [29], denoted by k and k , respectively. Hence,
n+1
n+1
w1 (n, k) =
, W1 (n, k) =
.
k+1
k+1
We note that Benoumhani [3, 4] already established the fundamental properties of the numbers wm (n, k) and Wm (n, k) while Dowling [20] gave a detailed
n discussion of geometric latn
tices. Other generalizations of the Stirling numbers k and k were already considered by
2
n+r
n+r
and \
several authors. For instance, Broder [5] defined the r-Stirling numbers [
k+r r
k+r 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
f(α) (n, k) of the first and second kind, which are related to the
numbers w
e(α) (n, k) and W
Stirling numbers via
n
n−k
n−k n
f(α) (n, k) = α
.
, W
w
e(α) (n, k) = α
k
k
Furthermore, Mező [27] defined the r-Whitney numbers wm,r (n, k) and Wm,r (n, k) of the first
and second kind as the coefficients in the expressions
n
m (x)n =
n
X
wm,r (n, k)(mx + r)k
(3)
k=0
and
n
(mx + r) =
n
X
mk Wm,r (n, k)(x)k .
(4)
k=0
respectively. Further developement of the numbers wm,r (n, k) and Wm,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
k
∆t (βt + γ|α)n t=0
,
(5)
Fα,γ (n, k) = lim
β→0
k!β k
where
n−1
Y
(βt + γ|α)n =
(βt + γ − jα) , (βt + γ|α)0 = 1,
(6)
j=0
which is a generalization of the Stirling numbers of the first kind. The numbers Fα,−γ (n, k)
are actually the r-Whitney numbers of the first kind in (3). That is,
Fα,−γ (n, k) = wα,γ (n, k).
Similarly, Corcino [11] defined the (r, β)-Stirling numbers
n
t =
n t−r X
β
k=0
k
n
k r,β
n
β k!
.
k r,β
k
as coefficients in
(7)
The numbers nk r,β are found to be equivalent to the r-Whitney numbers of the second kind
in (4). To be precise,
n
= Wβ,r (n, k).
k r,β
3
Corcino et al. [16], and Corcino and Aldema [12] further studied the numbers nk 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 {|si ; s √
= 0, 1, 2, . . .}, to be referred to as Fock
states, the relations a|si = s|s − 1i and a† |si = s + 1|s + 1i form a representation that
satisfies the commutation relation (8). The operator n̂ ≡ a† a, when acting on |si, yields
a† a|si = s|si,
and the operator (a† )k ak , when acting on the same state, yields
(a† )k ak |si = (s)k |si.
Let {hs| ≡ (|si)† ; s = 0, 1, 2, . . .} denote the Fock basis of the dual space. Requiring the
normalization of the scalar product h0|0i = 1 we note that
1
1 hs + 1|s + 1i =
hs|aa† |si =
hs|a† a|si + hs|si = hs|si.
s+1
s+1
Hence, from the normalization of√|0 > it follows that all the Fock states are
√ normalized.
†
†
Moreover, since hs + 1|a |si = s + 1hs + 1|s + 1i and (a|s + 1i) |si = s + 1hs|si, 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.
n Hence, the horizontal generating functions of the Stirling numbers k and nk ,
n
X
n−k n
(x)n =
xk
(9)
(−1)
k
k=0
and
n
x =
n X
n
k
k=0
can be expressed as
(x)k ,
n
X
n−k n
(a† a)k
(a ) a =
(−1)
k
k=0
† n n
and
n X
n
(a† )k ak ,
(a a) =
k
k=0
†
n
respectively [22].
Now, the defining relations for the r-Whitney numbers, (3) and (4), can be expressed as
n
† n n
m (a ) a =
n
X
wm,r (n, k)(ma† a + r)k
k=0
4
(10)
and
n
†
(ma a + r) =
n
X
mk Wm,r (n, k)(a† )k ak .
(11)
k=0
Making use of the q-Boson operators [1] that satisfy
[a, a† ]q ≡ aa† − qa† a = 1,
we have
a|si =
hence,
q
†
[s]q |s − 1i, a |si =
q
(12)
[s + 1]q |s + 1i,
a† a|si = [s]q |si,
and
(a† )k ak |si = [s]q,k |si.
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
relations for Carlitz’s [6] q-Stirling numbers of the first and
nthis, the
ndefining
second kind, k q and k q , can be written in the form [22]
† n n
(a ) a =
n
X
(−1)
n−k
k=1
and
†
n
(a a) =
n X
n
k
k=1
n
(a† a)k
k q
(13)
(a† )k ak ,
(14)
q
respectively.
We define q-analogues for the Whitney numbers wm,r (n, k) and Wm,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 wm,r,q (n, k) and Wm,r,q (n, k),
respectively, are defined by
mn (a† )n an =
n
X
wm,r,q (n, k)(ma† a + r)k
k=0
5
(15)
and
n
†
(ma a + r) =
n
X
mk Wm,r,q (n, k)(a† )k ak
(16)
k=0
with initial conditions wm,r,q (0, 0) = Wm,r,q (0, 0) = 1 and wm,r,q (n, k) = Wm,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),
n
,
(17)
wm,0,q (n, k) = (−m)
k q
n−k n
Wm,0,q (n, k) = m
.
(18)
k q
n+r
n+r
are specified by the horizontal generating
and \
Similarly, the r-Stirling numbers [
k+r r
k+r r
functions
\
n
X
n−k n + r
(x − r)n =
xk ,
(−1)
k+r r
k=0
n−k
or, equivalently,
and
\
n
X
n−k n + r
(x)n =
(x + r)k ,
(−1)
k
+
r
r
k=0
n
(x + r) =
n \
X
n+r
k=0
n+r
n+r
\
, the q-analogues of
and
Hence, [
k+r q,r
k+r q,r
by the horizontal generating functions
† n n
(a ) a
(a† a + r)n
It follows that
k+r
[
n+r
k+r r
(x)r .
r
n+r
, respectively, are specified
and \
k+r r
\
n+r
=
(a† a + r)k ,
(−1)
k + r q,r
k=0
n \
X
n+r
=
(a† )k ak .
k
+
r
q,r
k=0
n
X
n−k
\
n+r
w1,r,q (n, k) = (−1)
,
k + r q,r
\
n+r
W1,r,q (n, k) =
.
k + r q,r
n−k
(19)
(20)
(21)
(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.
6
Theorem 3. The (q, r)-Whitney numbers wm,r,q (n, k) and Wm,r,q (n, k) satisfy the following
identities:
n X
i i−k n−i n
n−k
r m
,
(23)
wm,r,q (n, k) = (−1)
i
k
q
i=k
n
X n
n−i i−k i
.
(24)
r m
Wm,r,q (n, k) =
k
i
q
i=k
Proof. From Eq. (13), we get
n
† n n
m (a ) a
=
=
=
=
n
m
(−1)
(a† a)i
i
q
i=0
i
n
X
ẑ − r
n−i n
n
(−1)
m
i q
m
i=0
i n
X
1 X i k
n
n−i n
ẑ (−r)i−k
m
(−1)
i
k
m
i
q
i=0
k=0
( n
)
n
X
X
n
i
(−1)n−k
ri−k ẑ k ,
mn−i
i
k
q
k=0
i=k
n
n
X
n−i
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
n X
n n−i i † i
†
n
(ma a + r) =
r m (a a)
i
i=0
n i X
n n−i i X i
=
r m
(a† )k ak
k q
i
i=0
k=0
(
)
n
n
X
X
i
n
rn−i mi
=
(a† )k ak .
k
i
q
i=k
k=0
Comparing the coefficient of (a† )k ak with that in equation (16) gives us (24).
Remark 4. (a) As q → 1, we have
n
X
i−k n−i n
n−k i
;
r m
wm,r (n, k) =
(−1)
i
k
i=k
n X
n n−i i−k i
.
r m
Wm,r (n, k) =
k
i
i=k
7
(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
\
n X
n+r
i i−k n
r
=
;
k + r q,r
i q
k
i=k
(25)
\
n X
n n−i i
n+r
r
.
=
k
i
k + r q,r
q
i=k
3
(26)
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 ]qn = [n]q (a† )n−1
and
[an , a† ]qn = [n]q an−1 ,
that can be easily established by induction. The latter can also be written in the form
a† an = q −n (an a† − [n]q an−1 ).
Theorem 6. The (q, r)-Whitney numbers wm,r,q (n, k) and Wm,r,q (n, k) satisfy the following
triangular recurrence relations:
wm,r,q (n + 1, k) = q −n wm,r,q (n, k − 1) − (m[n]q + r)wm,r,q (n, k) ,
(27)
Wm,r,q (n + 1, k) = q k−1 Wm,r,q (n, k − 1) + (m[k]q + r)Wm,r,q (n, k).
Proof. From equation (15),
n+1
X
(28)
wm,r,q (n + 1, k)(ma† a + r)k = mn+1 (a† )n (a† an )a
k=0
= mn+1 (a† )n q −n (an a† − [n]q an−1 )a
= mn+1 q −n (a† )n an (a† a) − mn+1 q −n [n]q (a† )n an
n
n
X
X
wm,r,q (n, k)(ma† a + r)k
= q −n
wm,r,q (n, k)(ma† a + r)k (ma† a + r − r) − mq −n [n]q
= q −n
= q −n
k=0
n+1
X
k=1
n+1
X
k=0
k=0
wm,r,q (n, k − 1)(ma† a + r)k − q −n (m[n]q + r)
n
X
wm,r,q (n, k)(ma† a + r)k
k=0
{wm,r,q (n, k − 1) − (m[n]q + r)wm,r,q (n, k)} (ma† a + r)k .
8
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 wm,r,q (n, k) and
Wm,r,q (n, k), using the initial values specified above.
Remark 7. (a) From (27) we obtain the explicit expression
wm,r,q (n, 0) = (−1)n q −
n−1
Y
n(n−1)
2
(m[i]q + r).
i=0
On the other hand, the relation (23) yields
wm,r,q (n, 0) = (−1)
n
n
X
i
rm
n−i
i=0
Equating these expressions and substituting x =
n X
n
i
i=0
i
x =q
−
r
m
n(n−1)
2
q
n
.
i q
we obtain
n−1
Y
([i]q + x).
i=0
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
n−1
Y
qi = q(2),
n
i=0
we obtain
n X
n
i=0
i
(−1)i [s]iq = (−1)n
q
n−1
Y
i=0
[s − i]q .
(b) From (28) Wm,r,q (n + 1, 0) = rWm,r,q (n, 0), hence Wm,r,q (n, 0) = rn . The same result is
obtained from (24). That is,
n X
n n−i i
r m δi,0 = rn .
Wm,r,q (n, 0) =
i
i=0
(c) As q → 1, we have
wm,r (n + 1, k) = wm,r (n, k − 1) − (mn + r)wm,r (n, k);
Wm,r (n + 1, k) = Wm,r (n, k − 1) + (mk + r)Wm,r (n, k).
This confirms that wm,r,q (n, k) and Wm,r,q (n, k) are proper q-analogues of wm,r (n, k) and
Wm,r (n, k), respectively.
9
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:
\
\ \ n+r
n+1+r
−n n + r
−n
,
+ ([n]q + r)q
= q
k + r q,r
k − 1 + r q,r
k + r q,r
\ \ \
n+1+r
n+r
n+r
k−1
= q
+ ([k]q + r)
.
k+r
k − 1 + r q,r
k + r q,r
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
n X
k
(−1)k−ℓ wm,r,q (n, k),
wm,r+1,q (n, ℓ) =
ℓ
k=ℓ
Wm,r+1,q (n, k) =
n X
n
ℓ=k
Proof. From equation (15), we have
n
† n n
m (a ) a
=
n
X
ℓ
Wm,r,q (ℓ, k).
wm,r,q (n, k)(ma† a + r)k
k=0
=
=
n
X
k=0
n
X
k
wm,r,q (n, k) (ma† a + r + 1) − 1
wm,r,q (n, k)
k=0
k X
k
ℓ=0
ℓ
(−1)k−ℓ (ma† a + r + 1)ℓ
n
n X
X
k
†
ℓ
(−1)k−ℓ wm,r,q (n, k).
=
(ma a + r + 1)
ℓ
ℓ=0
k=ℓ
On the other hand,
n
† n n
m (a ) a =
n
X
wm,r+1,q (n, ℓ)(ma† a + r + 1)ℓ .
ℓ=0
Hence, by comparing the coefficients of (ma† a + r + 1)ℓ we obtain equation (29).
Similarly, from equation (16)
†
n
(ma a + r + 1) =
n
X
mk Wm,r+1,q (n, k)(a† )k ak ,
k=0
10
(29)
(30)
and since
†
(ma a + r + 1)
n
=
n X
n
ℓ=0
n X
ℓ
(ma† a + r)ℓ
ℓ
n X k
m Wm,r,q (ℓ, k)(a† )k ak
=
ℓ
k=0
ℓ=0
n
n X
X
n
k † k k
Wm,r,q (ℓ, k),
=
m (a ) a
ℓ
k=0
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.
\ \
n
X
n+r+1
n+r
l−k k
=
(−1)
,
l + r + 1 q,r+1
l k + r q,r
k=l
\ n \
X
n
l+r
n+r+1
.
=
l
k
+
r
k + r + 1 q,r+1
q,r
l=k
4
Orthogonality and inverse relations
Theorem 11. The (q, r)-Whitney numbers wm,r,q (n, k) and Wm,r,q (k, j) satisfy the following
orthogonality relations:
n
X
Wm,r,q (n, k)wm,r,q (k, j) = δjn ,
(31)
k=j
and
n
X
wm,r,q (n, k)Wm,r,q (k, j) = δjn ,
(32)
k=j
where δjn is the Kronecker delta.
Proof. Using equation (15) we substitute mk (a† )k ak in (16), obtaining
†
(ma a + r)
n
=
n
X
k=0
=
n
X
j=0
Wm,r,q (n, k)
(
k
X
j=0
n
X
wm,r,q (k, j)(ma† a + r)j
)
Wm,r,q (n, k)wm,r,q (k, j) (ma† a + r)j .
k=j
11
Comparing the coefficients of (ma† a + r)j yields equation (31). Equation (32) is obtained
similarly.
The classical binomial inversion formula given by
fk =
k X
k
j
j=0
gj ⇔ gk =
k
X
(−1)
k−j
j=0
k
fj
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]
fk =
k X
k
j=0
j
q
gj ⇔ gk =
k
X
(−1)
k−j
j=0
k−j
k
(
)
2
q
fj ,
j q
(34)
The next theorem presents an inverse relation for the (q, r)-Whitney numbers wm,r,q (n, k)
and Wm,r,q (k, j).
Theorem 12. The (q, r)-Whitney numbers wm,r,q (n, ℓ) and Wm,r,q (n, ℓ) satisfy the following
inverse relation:
n
n
X
X
fn =
wm,r,q (n, ℓ)gℓ ⇔ gn =
Wm,r,q (n, ℓ)fℓ .
(35)
ℓ=0
ℓ=0
Proof. By the hypothesis,
n
X
Wm,r,q (n, ℓ)fℓ =
ℓ=0
=
=
n
X
Wm,r,q (n, ℓ)
ℓ=0
( n
n
X
X
k=0
n
X
k=0
ℓ
X
k=0
wm,r,q (ℓ, k)gk
)
Wm,r,q (n, ℓ)wm,r,q (ℓ, k) gk
ℓ=k
{δkn } gk
= gn .
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 wm,r,q (n, ℓ) and Wm,r,q (n, ℓ) satisfy the following
inverse relation:
∞
∞
X
X
fℓ =
wm,r,q (n, ℓ)gn ⇔ gℓ =
Wm,r,q (n, ℓ)fn .
(36)
n=ℓ
n=ℓ
12
5
(q, r)-Dowling polynomials and numbers
Cheon and Jung [7] defined the r-Dowling polynomials, denoted by Dm,r (n, x), in terms of
sums of r-Whitney numbers of the second kind. That is,
Dm,r (n, x) =
n
X
Wm,r (n, k)xk .
(37)
k=0
When x = 1, we obtain the r-Dowling numbers Dm,r (n) ≡ Dm,r (n, 1). The polynomials (37)
are actually equivalent to the (r, β)-Bell polynomials Gn,β,r (x) of R. B. Corcino and C. B.
Corcino [13]. That is,
Dβ,r (n, x) = Gn,β,r (x).
Moreover,
• when m = 1 and r = 1, we recover the classical Dowling polynomials D(n, x) ≡
D1,1 (n, x);
• when m = 1 and r = 0, we recover the classical Bell polynomials Bn (x) ≡ D1,0 (n, x);
• when m = 1, we recover Mező’s [28] r-Bell polynomials Bn,r (x). That is, D1,r (n, x) =
Bn,r (x); and
e (α) (n; x) by
• when m = α and r = 0, we recover the translated Dowling polynomials D
e (α) (n; x).
Mangontarum et al. [25]. That is, Dα,0 (n, x) = D
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 Dm,r,q (n, x), are defined by
Dm,r,q (n, x) =
n
X
Wm,r,q (n, k)xk
(38)
k=0
and the (q, r)-Dowling numbers, denoted by Dm,r,q (n), are defined by
The coherent states
Dm,r,q (n) = Dm,r,q (n, 1).
(39)
|γ|2 X γ k
√ |ki,
|γi = exp −
2
k!
k≥0
(40)
where γ is an arbitrary (complex-valued) constant, satisfy a|γi = γ|γi and hγ|γi = 1. Katriel
[23] gave an illustration on how (40) can be a very useful tool in the derivation of certain
13
Dobinski-type formulas. The q-coherent states corresponding to the q-Boson operators were
defined as
1 X γk
p
|ki
(41)
|γiq = ebq (−|γ|2 ) 2
[k]
!
q
k≥0
which satisfy a|γi = γ|γi. Here, ebq (x) is the type 2 q-exponential function given by
ebq (x) =
∞
Y
i=1
(1 + (1 − q)q i−1 x) =
X
i≥0
q (2)
i
xi
,
[i]q !
(42)
which is the inverse of the type 1 q-exponential function
eq (x) =
∞
Y
i=1
(1 − (1 − q)q i−1 x)−1 =
X xi
.
[i]
!
q
i≥0
(43)
That is, eq (x)b
eq (−x) = 1.
Taking the expectation value of both sides of (16) with respect to |γi yields
hγ|(ma† a + r)n |γi =
n
X
mk Wm,r,q (n, k)|γ|2k .
(44)
k=0
The left-hand-side can be evaluated using the q-coherent states in (41), yielding
hγ|(ma† a + r)n |γi = ebq −|γ|2
X |γ|2k
(m[k]q + r)n .
[k]
!
q
k≥0
(45)
Defining x = m|γ|2 we obtain
x X x k (m[k] + r)n
q
.
Wm,r,q (n, k)xk = ebq −
m k≥0 m
[k]q !
k=0
n
X
(46)
Using (38), the following theorem is easily observed.
Theorem 15. The (q, r)-Dowling polynomials Dm,r,q (n, x) and the (q, r)-Dowling numbers
Dm,r,q (n) have the following explicit formulas:
and
Dm,r,q (n, x) = ebq
x X x k (m[k]q + r)n
−
,
m k≥0 m
[k]q !
X (m[k]q + r)n
.
Dm,r,q (n) = ebq −m
mk [k]q !
k≥0
−1
14
(47)
(48)
Proof. (48) can be obtained by letting x = 1 in (47).
Katriel [23] defined the q-Bell polynomial as
k X
k
ℓ=0
ℓ
ℓ
q
x = ebq (x)
∞
X
xm
m=1
[m]kq
.
[m]q !
(49)
Expanding the right-hand side using (42) yields
k X
k
∞
ℓ
X
xℓ X
ℓ
ℓ−j (ℓ−j
x =
(−1) q 2 )
[j]kq .
ℓ q
[ℓ]q ! j=0
j q
ℓ=0
ℓ=0
ℓ
(50)
Equating coefficients of equal powers of x gives us
ℓ
ℓ−j
k
1 X
ℓ−j ( 2 ) ℓ
(−1) q
[j]k .
=
j q q
ℓ q [ℓ]q ! j=0
Notice that as q → 1, (51) reduces to the well-known explicit formula of
ℓ
1X
k
ℓ−j ℓ
=
lim
jk.
(−1)
q→1 ℓ
ℓ! j=0
j
q
(51)
k
j
. That is
(52)
In the following theorem, we will present an expression analogous to (51) for the q-analogue
Wm,r,q (n, k).
Theorem 16. The (q, r)-Whitney numbers of the second kind, Wm,r,q (n, k), have the following explicit formula:
ℓ
ℓ−k
1 X
ℓ−k ( 2 ) ℓ
Wm,r,q (n, ℓ) = ℓ
(−1) q
(m[k]q + r)n .
m [ℓ]q ! k=0
k q
Proof. Substituting y =
n
X
k=0
x
m
(53)
in (47) gives us
X
X
(m[k]q + r)n
[k]q !
i≥0
k≥0
ℓ
X yℓ X
ℓ−k
ℓ−k ( 2 ) ℓ
(−1) q
(m[k]q + r)n .
=
k
[ℓ]
q!
q
k=0
ℓ≥0
mk Wm,r,q (n, k)y k =
i (−y)
q (2)
[i]q !
i
yk
Equating the coefficients of equal powers of y on both sides of this equation we obtain
equation (53).
15
Note that as q → 1, we have
ℓ
1 X
ℓ−k ℓ
lim Wm,r,q (n, ℓ) =
(mk + r)n
(−1)
q→1
mℓ ℓ! k=0
k
= Wm,r (n, ℓ).
Furthermore,
lim Wm,1,q (n, l) = Wm (n, l).
q→1
Remark 17. We can also prove (53) in the following manner: First, we write (16) as
(m[ℓ]q + r)
n
=
=
n
X
mk Wm,r,q (n, k)[ℓ]q,k
k=0
ℓ X
k=0
ℓ
k
(
q
mk Wm,r,q (n, k)[ℓ]q,k
ℓ
k q
)
.
Next, we apply the q-binomial inversion formula in (34) which gives us
ℓ
ℓ−k
mℓ Wm,r,q (n, ℓ)[ℓ]q,ℓ X
ℓ−k ( 2 ) l
=
(−1) q
(m[k]q + r)n .
k
k
q
k
k=0
q
This is precisely the explicit formula obtained in the previous theorem.
Now, using (53),
k
n
X
XX
tn
k
(−1)k−j (k−j
n t
)
2
Wm,r,q (n, k)
=
q
(m[j]
+
r)
q
[n]q !
mk [k]q !
[n]q !
j q
n≥0
n≥0 j=0
k
X
k
1
k−j (k−j
(−1) q 2 )
eq [(m[j]q + r)t] ,
=
k
m [k]q ! j=0
j q
where eq (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
X
X
1
k
tn
k−j (k−j
)
2
= k
(−1) q
eq [(m[j]q + r)t] .
(54)
Wm,r,q (n, k)
[n]q !
m [k]q ! j=0
j q
n≥0
Remark 19. As q → 1, we have
X
ert
tn
=
lim
Wm,r,q (n, k)
q→1
[n]q !
k!
n≥0
emt − 1
m
k
,
which is the exponential generating function of the r-Whitney numbers of the second kind.
16
The q-difference operator [24] can be written in the form
k
X
k
k
k−j (k−j
)
∆q f (x) =
(−1) q 2
f (x + j).
j
q
j=0
(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:
X
eq [(m[x]q + r)t]
tn
k
= ∆q
.
(56)
Wm,r,q (n, k)
k [k] !
[n]
!
m
q
q
x=0
n≥0
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
(m[x]q + r)n
k
.
(57)
Wm,r,q (n, k) = ∆q
mk [k]q !
x=0
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.
P
For f (x) = k ck xk we obtain
X
a† af (a† a) =
ck (a† a)(a† a)k
k
=
X
ck a† (1 + qa† a)k a = a†
k
†
X
k
= a f (1 + qa† a)a.
17
ck (1 + qa† a)k
!
a
Remark 23. The lemma can also be written in the form
1 †
†
†
†
a g(a a)a = a ag
(a a − 1) ,
q
(59)
where g(x) is a polynomial.
Theorem 24 (Identity 1). The (q, r)-Whitney numbers of the second kind satisfy
n
X
n ℓ
q (m + r(1 − q))n−ℓ Wm,r,q (ℓ, k − 1).
Wm,r,q (n + 1, k) − rWm,r,q (n, k) =
ℓ
ℓ=k−1
Proof. In terms of the identity (58) and with the aid of (16)
n
a† m(1 + qa† a) + r a = a† a(ma† a + r)n
1
(ma† a + r − r)(ma† a + r)n
m
1
r
=
(ma† a + r)n+1 − (ma† a + r)n
m
m
n+1
X
=
mk−1 (a† )k ak Wm,r,q (n + 1, k) − rWm,r,q (n, k) .
=
k=0
On the other hand, defining α = m + r(1 − q) (which will hold throught the present section),
n
n
a† m(1 + qa† a) + r a = a† q(ma† a + r) + α a
!
n X
n
q ℓ αn−ℓ (ma† a + r)ℓ a
= a†
ℓ
ℓ=0
n ℓ
X
n ℓ n−ℓ X k
=
qα
m Wm,r,q (ℓ, k)(a† )k+1 ak+1
ℓ
ℓ=0
k=0
n+1
n
X
X
n ℓ n−ℓ
k−1 † k k
q α Wm,r,q (ℓ, k − 1).
=
m (a ) a
ℓ
k=1
ℓ=k−1
Equating coefficients of mk−1 (a† )k ak the theorem follows.
For r = 0 this identity reduces to the q-Stirling numbers identity [22, identity 1]
n
X
n ℓ n−ℓ
q m Wm,0,q (ℓ, k − 1).
Wm,0,q (n + 1, k) =
ℓ
ℓ=k−1
The following corollary is an immediate consequence of the previous theorem.
18
Corollary 25. As q → 1,
n
X
n
mn−ℓ Wm,r (ℓ, k − 1).
Wm,r (n + 1, k) − rWm,r (n, k) =
ℓ
ℓ=k−1
Theorem 26 (Identity 2). The (q, r)-Whitney numbers of the first kind satisfy
n
k−ℓ
X
1
wm,r,q (n + 1, ℓ) =
wm,r,q (n, k) − (m + r(1 − q)
qk
k=ℓ−1
k
k
·
(−(m + r(1 − q))) − r
.
ℓ−1
ℓ
Proof. We note that from (15),
m
n+1
† n+1 n+1
(a )
a
=
n+1
X
wm,r,q (n + 1, ℓ)(ma† a + r)ℓ .
ℓ=0
On the other hand, using (59),
mn+1 (a† )n+1 an+1 = ma† mn (a† )n an a
= ma†
n
X
k=0
= ma† a
n
X
wm,r,q (n, k)(ma† a + r)k a
wm,r,q (n, k)
m
(a† a − 1) + r
k
q
k
1
†
†
= ma a
wm,r,q (n, k) k (ma a + r) − α
q
k=0
n
k X
1 X k
†
(ma† a + r)ℓ (−α)k−ℓ
= ((ma a + r) − r)
wm,r,q (n, k) k
ℓ
q
k=0
ℓ=0
n
k
X
1 X k
=
(ma† a + r)ℓ+1 (−α)k−ℓ
wm,r,q (n, k) k
ℓ
q
k=0
ℓ=0
n n
X
1 X k
(ma† a + r)ℓ (−α)k−ℓ
−r
wm,r,q (n, k) k
ℓ
q
ℓ=0
k=0
k=0
n
X
n+1
X
n
X
1
=
(ma a + r)
w
(n, k)(−α)k−ℓ ·
k m,r,q
q
ℓ=0
k=ℓ−1
k
k
.
(−α) − r
·
ℓ
ℓ−1
†
ℓ
Equating the coefficients of equal powers of ma† a + r we obtain the theorem.
19
For r = 0, we recover the q-Stirling numbers identity [22, identity 2]
n
X
k
1
k−ℓ+1
w
(n, k)(−m)
,
wm,0,q (n + 1, ℓ) =
k m,0,q
ℓ
−
1
q
k=ℓ−1
Moreover, we have the following corollary:
Corollary 27. As q → 1,
wm,r (n + 1, ℓ) = −
n
X
wm,r (n, k)(−m)
k=ℓ−1
k−ℓ
k
k
.
+r
m
ℓ
ℓ−1
Theorem 28 (Identity 3). The (q, r)-Whitney numbers of the second kind satisfy
n+1
1 X
n
n
n−ℓ
Wm,r,q (n, k−1) = n
(−m − r(1 − q)) −
r Wm,r,q (ℓ, k).
(−m−r(1−q))
q ℓ=k
ℓ−1
ℓ
Proof. Note that
†
†
n
a (ma a + r) a =
=
n
X
k=0
n+1
X
k=1
mk Wm,r,q (n, k)(a† )k+1 ak+1
mk−1 Wm,r,q (n, k − 1)(a† )k ak ,
and on the other hand, using (59),
n
m
1
(a† a − 1) + r = a† a n (ma(† a + r) − α)n
a† (ma† a + r)n a = a† a
q
q
n
1
1 X n
=
(ma† a + r)ℓ (−α))n−ℓ
((ma† a + r) − r) n
m
q ℓ=0 ℓ
n+1
1 X
n
n
†
ℓ
n−ℓ
=
(−α) − r
(ma a + r) (−α)
·
n
ℓ
ℓ−1
mq ℓ=1
n+1
n+1
n
n
1 X k † k kX
n−ℓ
(−α) −
r Wm,r,q (ℓ, k).
m (a ) a
(−α)
=
ℓ−1
ℓ
mq n k=0
ℓ=k
Equating the coefficients of (a† )k ak we obtain the theorem.
For r = 0 this theorem reduces to
n+1
n
1 X
n+1−ℓ
Wm,0,q (ℓ, k).
(−m)
Wm,0,q (n, k − 1) = n
ℓ−1
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.
20
Corollary 29. As q → 1,
Wm,r (n, k − 1) =
n+1
X
(−m)
n−ℓ
ℓ=k
n
n
r Wm,r (ℓ, k).
(−m) −
ℓ
ℓ−1
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 Fα,γ (n, k) in (5), denoted by φα,γ [n, k]q . φα,γ [n, k]q can
be defined in terms of the relation
n
X
k=0
φα,γ [n, k]q tk = ht + [γ]q |[α]q iqn ,
where
ht +
[γ]q |[α]q iqn
=
n−1
Y
j=0
(t + [γ]q − [jα]q ) .
(60)
(61)
The numbers φα,γ [n, k]q are actually q-analogues of the numbers wm,r (n, k). Similarly, Corcino and Montero [18] defined the q-analogue σ[n, k]β,r
of the Rucinski-Voigt numbers in
q
terms of the reccurence relation
β,r
β,r
σ[n, k]β,r
q = σ[n − 1, k − 1]q + ([kβ]q + [r]q ) σ[n − 1, k]q .
(62)
σ[n, k]β,r
is also a q-analogue of the numbers nk r,β and Wm,r (n, k). However, by comparing
q
the defining relations for φα,γ [n, k]q and σ[n, k]β,r
with those of the(q, r)-Whitney numbers
q
wm,r,q (n, k) and Wm,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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[27] I. Mező, A new formula for the Bernoulli polynomials, Results Math., 58 (2010), 329–
335.
[28] I. Mező, The r-Bell Numbers, J. Integer Seq., 14 (2011), Article 11.1.1.
[29] J. Stirling, Methodus Differentialissme Tractus de Summatione et Interpolatione Serierum Infinitarum, London, 1730.
2010 Mathematics Subject Classification: Primary 11B83; Secondary 11B73, 05A30.
Keywords: Whitney number, Stirling number, Boson operator, q-analogue.
(Concerned with sequences A000110, A003575, A008275, and A008277.)
Received May 13 2015; revised version received, September 1 2015. Published in Journal of
Integer Sequences, September 7 2015.
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