# The Hankel Transform ofq-Noncentral Bell Numbers

Authors: Cristina B. Corcino; Roberto B. Corcino; Jay M. Ontolan; Charrymae M. Perez-Fernandez; Ednelyn R. Cantallopez
Journal: International Journal of Mathematics and Mathematical Sciences
 (2015)

---

## Abstract

We define two forms ofq-analogue of noncentral Stirling numbers of the second kind and obtain some properties parallel to those of noncentral Stirling numbers. Certain combinatorial interpretation is given for the second form of the q-analogue in the context of 0-1 tableaux which, consequently, yields certain additive identity and some convolution-type formulas. Finally, a q-analogue of noncentral Bell numbers is defined and its Hankel transform is established.


---

## Body

## 1. Introduction

TheHankel matrix
H
n of order n of a sequence A
=
{
a
0
,
a
1
,
…
,
a
n
} is given by H
n
=
(
a
i
+
j
)
0
≤
i
,
j
≤
n. TheHankel determinant
h
n of order n of A is the determinant of the corresponding Hankel matrix of order n. That is, h
n
=
det
⁡
(
H
n
). TheHankel transform of the sequence A, denoted by H
(
A
), is the sequence {
h
n
} of Hankel determinants of A. For instance, the Hankel transform of the sequence of Catalan numbers, C
=
{
(
1
/
(
n
+
1
)
)
2
n
n
}
n
=
1
∞, is given by(1)
H
C
=
1,1
,
1
,
…and the sequence of the sum of two consecutive Catalan numbers, a
n
=
c
n
+
c
n
+
1, with c
n being the nth Catalan numbers, has the Hankel transform(2)
H
a
n
=
F
2
n
+
1
n
=
0
∞
,where F
n is the nth Fibonacci number [1].One remarkable property of Hankel transform is established by Layman [1], which states that the Hankel transform of an integer sequence is invariant under binomial and inverse binomial transforms. We recall that a sequence B
=
(
b
n
) is a binomial transform of A
=
(
a
n
), denoted by B
(
A
), if(3)
b
n
=
∑
k
=
0
n
n
k
a
kand C
=
(
c
n
) is an inverse binomial transform of A
=
(
a
n
), denoted by C
(
A
), if(4)
c
n
=
∑
k
=
0
n
-
1
n
-
k
n
k
a
k
.In fact, Riordan [2, 3] established the so-calledbinomial inversion formula which is given by (5)
f
n
=
∑
k
=
0
n
n
k
g
k
⟺
g
n
=
∑
k
=
0
n
-
1
n
-
k
n
k
f
k
.Hence, if A is an integer sequence, B is binomial transform of A, and C is the inverse binomial transform of A, then (6)
H
B
A
=
H
A
,
H
C
A
=
H
A
.This property played an important role in proving that the Hankel transform of the sequence of ordinary Bell number {
B
n
} [4] and that of r-Bell numbers {
B
r
(
n
)
} [5] are equal. Recently, in the paper by R. B. Corcino and C. B. Corcino [6], this property has also been used in proving that the Hankel transform of the sequence of generalized Bell numbers {
G
n
,
r
,
β
} is given by(7)
H
G
n
,
r
,
β
=
∏
j
=
0
n
β
j
j
!
,where G
n
,
r
,
β is the sum of (
r
,
β
)-Stirling numbers n
k
r
,
β
(8)
G
n
,
r
,
β
=
∑
k
=
0
n
n
k
r
,
β (see [7, 8]).In that same paper [6], an open problem is disclosed which is to determine the Hankel transform of a q-analogue of generalized Bell numbers. This problem motivates the present authors to define q-analogues of noncentral Stirling and noncentral Bell numbers and, eventually, establish the Hankel transform of a q-analogue of noncentral Bell numbers. Results in this paper may be useful in proving the said open problem.


## 2. Aq-Analogue of S
a
(
n
,
k
): First Form

The noncentral Stirling numbers of the first and second kind [9], denoted by s
a
(
n
,
k
) and S
a
(
n
,
k
), respectively, are defined by means of the following inverse relations:(9)
t
n
=
∑
k
=
0
n
1
k
!
d
k
d
t
k
t
v
t
=
a
t
-
a
k
,
t
-
a
n
=
∑
k
=
0
n
1
k
!
Δ
k
t
-
a
n
t
=
0
t
k
,where a, t are any real numbers, n is a nonnegative integer, and (10)
s
a
n
,
k
=
1
k
!
d
k
d
t
k
t
v
t
=
a
(11)
S
a
n
,
k
=
1
k
!
Δ
k
t
-
a
n
t
=
0
.These numbers are a certain generalization of the classical Stirling numbers of the first and second kind, respectively, which satisfy the following recurrence relations: (12)
s
a
n
+
1
,
k
=
s
a
n
,
k
-
1
+
a
-
n
s
a
n
,
k
,
S
a
n
+
1
,
k
=
S
a
n
,
k
-
1
+
k
-
a
S
a
n
,
k
,with initial conditions (13)
s
a
0,0
=
1
,
s
a
n
,
0
=
a
n
,
s
a
0
,
k
=
0
,
n
,
k
≠
0
,
S
a
0,0
=
1
,
S
a
n
,
0
=
-
a
n
,
S
a
0
,
k
=
0
,
n
,
k
≠
0
.For a more detailed discussion of noncentral Stirling numbers, one may see [9]. Parallel to the ordinary Bell numbers, the noncentral Bell numbers can also be introduced as the sum of noncentral Stirling numbers of the second kind as follows: (14)
B
n
,
a
=
∑
k
=
0
n
S
a
n
,
k
.These numbers are equivalent to r-Bell numbers with the following relation: (15)
B
n
,
a
=
B
-
a
n
.Several mathematicians developed a way of obtaining a generalization of some special numbers. One generalization is aq-analogue of these special numbers. A q-analogue is a mathematical expression parameterized by a quantity q that generalizes a known expression and reduces to the known expression in the limit, as q
→
1. For instance, a polynomial a
k
(
q
) is a q-analogue of the polynomial a
k if lim
q
→
1
⁡
a
k
(
q
)
=
a
k. The q-analogues of n, n
!, (
n
)
k, and n
k are, respectively, given by (16)
n
q
=
1
+
q
+
q
2
+
⋯
+
q
n
-
1
=
1
-
q
n
1
-
q
;
n
q
!
=
n
q
n
-
1
q
⋯
2
q
1
q
;
n
k
,
q
=
n
q
n
-
1
q
⋯
n
-
k
+
1
q
;
n
k
q
=
n
q
!
k
q
!
n
-
k
q
!
=
n
k
,
q
k
!
.The polynomials n
k
q are usually called the q-binomial coefficients. These polynomials possess several properties including the q-binomial inversion formula (17)
f
n
=
∑
k
=
0
n
n
k
q
g
k
.
⟺
g
n
=
∑
k
=
0
n
-
1
n
-
k
q
n
-
k
2
n
k
q
f
k
,and certain generating function (18)
∑
k
=
0
n
q
k
2
n
k
q
x
k
=
1
+
x
1
+
x
q
1
+
x
q
2
⋯
1
+
x
q
n
-
1
.The q-binomial inversion formula is a q-analogue of the binomial inversion formula in (5).Aq-analogue of both kinds of Stirling numbers was first defined by Carlitz in [10], the second kind of which, known as q-Stirling numbers of the second kind, is defined in terms of the following recurrence relation:(19)
S
q
n
,
k
=
S
q
n
-
1
,
k
-
1
+
k
q
S
q
n
-
1
,
kin connection with a problem in Abelian groups, such that when q
→
1, this gives the triangular recurrence relation for the classical Stirling numbers of the second kind S
(
n
,
k
): (20)
S
n
,
k
=
S
n
-
1
,
k
-
1
+
k
S
n
-
1
,
k
.A different way of defining a q-analogue of Stirling numbers of the second kind has been adapted in [11] which is given as follows:(21)
S
q
n
,
k
=
q
k
-
1
S
q
n
-
1
,
k
-
1
+
k
q
S
q
n
-
1
,
k
.This type of q-analogue gives the Hankel transform of q-exponential polynomials and numbers which are a certain q-analogue of Bell polynomials and numbers. In the desire to establish the Hankel transform of q-analogue of generalized Bell numbers, the present authors are motivated to define a q-analogue of S
a
(
n
,
k
) parallel to that of (21) as follows.Definition 1.
For nonnegative integersn and k and real number a, a q-analogue S
a
[
n
,
k
]
q of S
a
(
n
,
k
) is defined by(22)
S
a
n
,
k
q
=
q
k
-
1
-
a
S
a
n
-
1
,
k
-
1
q
+
k
-
a
q
S
a
n
-
1
,
k
q
,where S
a
[
0,0
]
q
=
1, S
a
[
n
,
k
]
q
=
0 for n
<
k or n
,
k
<
0 and [
k
-
a
]
q
=
(
1
/
q
a
)
(
[
k
]
q
-
[
a
]
q
).Remark 2.
Whenq
→
1, the above definition will reduce to the recurrence relation of the noncentral Stirling numbers of the second kind established by Koutras [9] which is given by (23)
S
a
n
,
k
=
S
a
n
-
1
,
k
-
1
+
k
-
a
S
a
n
-
1
,
k
.Remark 3.
It can easily be verified that(24)
S
a
n
,
0
q
=
-
a
q
n
=
-
q
-
n
a
a
q
n
.By proper application of (22), we can easily obtain two other forms of recurrence relations and certain generating function.Theorem 4.
For nonnegative integersn and k and real number a, the q-analogue S
a
[
n
,
k
]
q satisfies the following vertical and horizontal recurrence relations: (25)
S
a
n
+
1
,
k
+
1
q
=
q
k
-
a
∑
j
=
k
n
k
+
1
-
a
q
n
-
j
S
a
j
,
k
q
,
(26)
S
a
n
,
k
q
=
∑
j
=
0
n
-
k
-
1
j
q
a
-
k
-
j
a
k
+
j
+
1
,
q
a
k
+
1
,
q
S
a
n
+
1
,
k
+
j
+
1
q
,respectively, where (27)
a
i
,
q
=
∏
h
=
1
i
-
1
1
q
h
-
1
h
q
-
a
q
=
∏
h
=
1
j
-
1
q
a
-
h
+
1
h
-
a
qwith initial value S
a
[
0,0
]
q
=
1. Moreover, a horizontal generating function for S
a
[
n
,
k
]
q is given by (28)
∑
k
=
0
n
S
a
n
,
k
q
t
+
a
k
,
q
=
t
q
n
.Explicit formulas and generating functions of a given sequence of numbers or polynomials are useful tools in giving combinatorial interpretation of the numbers or polynomials. In the subsequent theorems, we establish the exponential and rational generating functions and two explicit formulas forS
a
[
n
,
k
]
q. One of these explicit formulas is in symmetric function form which will be used to give combinatorial interpretation of S
a
[
n
,
k
]
q in the context of some A-tableaux.Aq-analogue of the difference operator, known as q-difference operator, was defined and thoroughly discussed in [12, 13]. More precisely, the q
-difference operator of degree n, denoted by Δ
q
,
h
n, is defined to be a mapping that assigns to every function f the function Δ
q
,
h
n
f defined by the rule(29)
Δ
q
,
h
n
f
x
=
∏
j
=
0
n
-
1
E
h
-
q
j
f
x
,
n
≥
1
,where E
h is the shift operator defined by E
h
f
(
x
)
=
f
(
x
+
h
). When h
=
1, we use the notation(30)
Δ
q
,
h
n
=
Δ
q
n
.As convention, define Δ
q
,
h
0
=
1 (the identity map). The following is the explicit formula for the q-difference operator:(31)
Δ
q
,
h
k
f
x
=
∑
j
=
0
k
-
1
k
-
j
q
k
-
j
2
k
j
q
f
x
+
j
h
.The newq-analogue of Newton’s Interpolation Formula in [13] states that, for (32)
f
q
x
=
a
0
+
a
1
x
-
x
0
q
+
⋯
+
a
m
x
-
x
0
q
x
-
x
1
q
⋯
x
-
x
m
-
1
q
,we have(33)
f
q
x
=
f
q
x
0
+
Δ
q
h
,
h
f
q
x
0
x
-
x
0
q
1
q
h
!
h
q
+
Δ
q
h
,
h
2
f
q
x
0
x
-
x
0
q
x
-
x
1
q
2
q
h
!
h
q
2
+
⋯
+
Δ
q
h
,
h
m
f
q
x
0
x
-
x
0
q
x
-
x
1
q
⋯
x
-
x
m
-
1
q
m
q
h
!
h
q
m
,where x
k
=
x
0
+
k
h, k
=
1,2
,
…, such that when x
0
=
0 and h
=
1, this can be simplified as(34)
f
q
x
=
f
q
0
+
Δ
q
f
q
0
x
q
1
q
!
+
Δ
q
2
f
q
0
x
q
x
-
1
q
2
q
!
+
⋯
+
Δ
q
m
f
q
0
x
q
x
-
1
q
⋯
x
-
m
-
1
q
m
q
!
.Using (28) with t
=
x, we get(35)
∑
k
=
0
n
S
a
n
,
k
x
+
a
k
,
q
=
x
q
n
,which can be expressed further as(36)
∑
k
=
0
n
S
a
n
,
k
x
q
x
-
1
q
⋯
x
-
k
-
1
q
=
x
-
a
q
n
.Applying the above Newton Interpolation Formula and the identity in (31) with h
=
1, (37)
Δ
q
k
f
x
=
∑
j
=
0
k
-
1
k
-
j
q
k
-
j
2
k
j
q
f
x
+
j
,we get(38)
S
a
n
,
k
=
Δ
q
k
f
q
0
k
q
!
=
1
k
q
!
∑
j
=
0
k
-
1
k
-
j
q
k
-
j
2
k
j
q
j
-
a
q
n
.Thus, we obtain the following explicit formula.Theorem 5.
The explicit formula forS
a
[
n
,
k
]
q is given by (39)
S
a
n
,
k
q
=
1
k
q
!
∑
j
=
0
k
-
1
k
-
j
q
k
-
j
2
k
j
q
j
-
a
q
n
.Remark 6.
The above theorem reduces to(40)
S
a
n
,
k
=
1
k
!
∑
j
=
0
k
-
1
k
-
j
k
j
j
-
a
nwhen q
→
1 which is the explicit formula of the noncentral Stirling numbers of the second kind.Remark 7.
For brevity, (39) can be expressed as (41)
S
a
n
,
k
q
=
1
k
q
!
Δ
q
k
x
-
a
q
n
x
=
0which reduces to (11) as q
→
1.Theorem 8.
For nonnegative integersn and k and real number a, the q-analogue S
a
[
n
,
k
]
q has a generating function (42)
∑
n
≥
0
S
a
n
,
k
q
t
q
n
n
q
!
=
1
k
q
!
Δ
q
k
e
q
x
-
a
q
t
q
x
=
0
.Proof.
Using the explicit formula in Theorem5, we obtain (43)
∑
n
≥
0
S
a
n
,
k
q
t
q
n
n
q
!
=
∑
n
≥
0
1
k
q
!
∑
j
=
0
k
-
1
k
-
j
q
k
-
j
2
k
j
q
j
-
a
q
n
t
q
n
n
q
!
=
1
k
q
!
∑
j
=
0
k
-
1
k
-
j
q
k
-
j
2
k
j
q
∑
n
≥
0
j
-
a
q
t
q
n
n
q
!
=
1
k
q
!
∑
j
=
0
k
-
1
k
-
j
q
k
-
j
2
k
j
q
e
q
j
-
a
q
t
q
.Using (31), we prove the theorem.Remark 9.
Whenq
→
1, the above theorem becomes(44)
1
k
!
∑
j
=
0
k
-
1
k
-
j
k
j
e
j
-
a
t
=
e
-
a
t
1
k
!
∑
j
=
0
k
-
1
k
-
j
k
j
e
t
j
=
e
-
a
t
1
k
!
e
t
-
1
kwhich is the exponential generating function of the noncentral Stirling numbers of the second kind.The following theorem contains the rational generating function forS
a
[
n
,
k
]
q.Theorem 10.
For nonnegative integersn and k and complex number a, the q-analogue S
a
[
n
,
k
]
q satisfies the rational generating function(45)
Ψ
k
t
=
∑
n
≥
k
S
a
n
,
k
q
t
q
n
=
q
k
2
-
k
a
t
q
k
∏
j
=
0
k
1
-
j
-
a
q
t
q
.Proof.
Fork
=
0, we have(46)
Ψ
0
t
=
∑
n
≥
0
S
a
n
,
0
q
t
q
n
=
∑
n
≥
0
-
a
q
n
t
q
n
=
1
1
-
-
a
q
t
q
.With k
>
0 and using Definition 1, we obtain(47)
Ψ
k
t
=
∑
n
≥
k
S
a
n
,
k
q
t
q
n
=
q
k
-
1
-
a
t
q
∑
n
≥
k
S
a
n
-
1
,
k
-
1
q
t
q
n
-
1
+
k
-
a
q
t
q
∑
n
≥
k
S
a
n
-
1
,
k
q
t
q
n
-
1
=
q
k
-
1
-
a
t
q
Ψ
k
-
1
t
+
k
-
a
q
t
q
Ψ
k
t
.Hence, (48)
Ψ
k
t
=
q
k
-
1
-
a
t
q
1
-
k
-
a
q
t
q
Ψ
k
-
1
t
,
Ψ
k
t
=
q
k
-
1
-
a
t
q
1
-
k
-
a
q
t
q
q
k
-
2
-
a
t
q
1
-
k
-
1
-
a
q
t
q
Ψ
k
-
2
t
⋮
Ψ
k
t
=
q
k
2
-
k
a
t
q
k
∏
j
=
0
k
1
-
j
-
a
q
t
q
.The rational generating function in Theorem10 can then be expressed as (49)
∑
n
≥
k
S
a
n
,
k
q
t
q
n
=
q
k
2
-
k
a
t
q
k
∏
j
=
0
k
1
1
-
j
-
a
q
t
q
=
q
k
2
-
k
a
t
q
k
∏
j
=
0
k
∑
n
≥
0
j
-
a
q
n
t
q
n
=
q
k
2
-
k
a
t
q
k
∑
n
≥
k
∑
s
1
+
s
2
+
⋯
+
s
k
=
n
-
k
∏
j
=
0
k
j
-
a
q
s
j
t
q
s
j
=
q
k
2
-
k
a
t
q
k
∑
n
≥
k
∑
s
1
+
s
2
+
⋯
+
s
k
=
n
-
k
∏
j
=
0
k
j
-
a
q
s
j
t
q
n
-
k
=
q
k
2
-
k
a
∑
n
≥
k
∑
s
1
+
s
2
+
⋯
+
s
k
=
n
-
k
∏
j
=
0
k
j
-
a
q
s
j
t
q
n
.Hence,(50)
S
a
n
,
k
q
=
q
k
2
-
k
a
∑
s
1
+
s
2
+
⋯
+
s
k
=
n
-
k
∏
j
=
0
k
j
-
a
q
s
j
.This sum may be written further as follows.Theorem 11.
For nonnegative integersn and k and complex number a, the explicit formula for S
a
[
n
,
k
]
q in homogeneous symmetric function form is given by(51)
S
a
n
,
k
q
=
∑
0
≤
j
1
≤
j
2
≤
⋯
≤
j
n
-
k
≤
k
q
k
2
-
k
a
∏
i
=
1
n
-
k
j
i
-
a
q
.


## 3. Aq-Analogue of S
a
(
n
,
k
): Second Form

We now define anotherq-analogue of the noncentral Stirling numbers, denoted by S
a
∗
[
n
,
k
]
q, as follows:(52)
S
a
∗
n
,
k
q
≔
q
k
a
-
k
2
S
a
n
,
k
q
.Hence,(53)
S
a
∗
n
,
k
q
=
∑
0
≤
j
1
≤
j
2
≤
⋯
≤
j
n
-
k
≤
k
∏
i
=
1
n
-
k
j
i
-
a
q
.All other properties parallel to those of S
a
[
n
,
k
]
q can easily be established by imbedding the factor q
k
a
-
k
2 in the derivations or multiplying this factor directly by both sides of the equation that yields the formula/identity for S
a
[
n
,
k
]
q.The following definition contains the concept ofA-tableau which is useful in describing S
a
∗
[
n
,
k
]
q, combinatorially.Definition 12 (see [14]).
AnA
-tableau is a list ϕ of columns c of Ferrer’s diagram of a partition λ (by decreasing order of length) such that the lengths |
c
| are part of the sequence A
=
(
a
i
)
i
≥
0, a strictly increasing sequence of nonnegative integers.Letω be a function from the set of nonnegative integers N to a ring K. Suppose Φ is an A-tableau with r columns of lengths |
c
|
≤
h. We use T
a
A
(
h
,
r
) to denote the set of such A-tableaux. Then, we set(54)
ω
A
Φ
=
∏
c
∈
Φ
ω
c
.Note that Φ might contain a finite number of columns whose lengths are zero since 0
∈
A
=
{
0,1
,
2
,
…
,
k
} and if ω
(
0
)
≠
0.From this point onward, whenever anA-tableau is mentioned, it is always associated with the sequence A
=
{
0,1
,
2
,
…
,
k
}.We are now ready to mention the following theorem.Theorem 13.
Letω
:
N
→
K denote a function from N to a ring K (column weights according to length) which is defined by ω
(
|
c
|
)
=
[
|
c
|
-
a
]
q, where a is a complex number and |
c
| is the length of column c of an A-tableau in T
a
A
(
k
,
n
-
k
). Then, (55)
S
a
∗
n
,
k
q
=
∑
ϕ
∈
T
a
A
k
,
n
-
k
∏
c
∈
ϕ
ω
c
.Proof.
This can easily be proved using Definition12 and (53).


### 3.1. Combinatorics ofA-Tableaux

Suppose that, for some numbersa
1 and a
2, we have a
=
a
1
+
a
2. Then, (53) yields (56)
S
a
∗
n
,
k
q
=
∑
0
≤
j
1
≤
j
2
≤
⋯
≤
j
n
-
k
≤
k
∏
i
=
1
n
-
k
j
i
-
a
1
-
a
2
q
.That is, for any ϕ
∈
T
a
A
(
k
,
n
-
k
), (57)
ω
A
ϕ
=
∏
c
∈
ϕ
j
i
-
a
1
-
a
2
q
,where |
c
|
∈
{
0,1
,
2
,
…
,
k
}. Note that the weight of each column of ϕ can be considered as a finite sum with additive constant a
2; that is, for each c
∈
ϕ, we can write(58)
ω
c
=
1
q
a
2
ω
∗
c
-
a
2
q
,where ω
∗
(
|
c
|
)
=
[
|
c
|
-
a
1
]
q. The following theorem determines how an additive constant affects the recurrence formula for S
a
[
n
,
k
]
q.From Theorem13,(59)
S
a
∗
n
,
k
q
=
∑
ϕ
∈
T
a
A
k
,
n
-
k
ω
A
ϕ
=
∑
ϕ
∈
T
a
A
k
,
n
-
k
∏
c
∈
ϕ
ω
c
,where (60)
ω
A
ϕ
=
∏
c
∈
ϕ
c
-
a
q
,
where
c
∈
0,1
,
…
,
k
=
∏
i
=
1
n
-
k
j
i
-
a
q
,
where
j
i
∈
0,1
,
…
,
k
.If a
=
a
1
+
a
2 for some a
1 and a
2, then, by (58), (61)
ω
A
ϕ
=
∏
i
=
1
n
-
k
1
q
a
2
ω
∗
j
i
-
a
2
q
,
where
ω
∗
j
i
=
j
i
-
a
1
q
=
q
-
n
-
k
a
2
ω
∗
j
1
-
a
2
q
ω
∗
j
2
-
a
2
q
⋯
ω
∗
j
n
-
k
-
a
2
q
=
q
-
n
-
k
a
2
∑
l
=
0
n
-
k
-
a
2
q
n
-
k
-
l
∑
j
1
≤
q
1
≤
q
2
≤
⋯
≤
q
l
≤
j
n
-
k
∏
i
=
1
l
ω
∗
q
i
.SupposeB
ϕ is the set of all A-tableaux corresponding to ϕ such that, for each ψ
∈
B
ϕ, either(62)
ψ
has  no  column  whose  weight  is
-
a
2
q
,
or
ψ
has  one  column  whose  weight  is
-
a
2
q
,
or
ψ
has  two  columns  whose  weights  are
-
a
2
q
,
or
⋮
ψ
has
n
-
k
columns  whose  weights  are
-
a
2
q
.Then, we may write (63)
ω
A
ϕ
=
∑
ψ
∈
B
ϕ
ω
A
ψ
.Now, if r columns in ψ have weights other than -
[
a
2
]
q, then(64)
ω
A
ψ
=
∏
c
∈
ψ
ω
∗
c
=
q
-
n
-
k
a
2
-
a
2
q
n
-
k
-
r
∏
i
=
1
r
ω
∗
q
i
,where q
1
,
q
2
,
…
,
q
r
∈
{
j
1
,
j
2
,
…
,
j
n
-
k
}. Note that, for each r, there correspond (65)
n
-
k
rtableaux with r columns having weights ω
∗
(
j
i
)
=
[
j
i
-
a
1
]
q. It can easily be verified that (66)
T
a
A
k
,
n
-
k
=
n
-
k
+
k
n
-
k
=
n
n
-
k
=
n
k
.Thus, ∀
ϕ
∈
T
a
A
(
k
,
n
-
k
), B
ϕ contains a total of (67)
n
k
n
-
k
rtableaux with r columns of weights ω
∗
(
j
i
). However, only (68)
r
+
k
rtableaux with r columns in B
ϕ are distinct. Hence, every distinct tableau ψ with r columns of weights other than -
[
a
2
]
q appears(69)
n
k
n
-
k
r
r
+
k
r
=
n
r
+
ktimes in the collection. Thus, (70)
∑
ϕ
∈
T
a
A
k
,
n
-
k
ω
A
ϕ
=
∑
r
=
0
n
-
k
n
r
+
k
q
-
n
-
k
a
2
-
a
2
q
n
-
k
-
r
∑
φ
∈
B
-
r
∏
c
∈
φ
ω
∗
c
,where B
-
r denotes the set of all tableaux φ having r columns of weights ω
∗
(
j
i
)
=
[
j
i
-
a
1
]
q. Reindexing the double sum, we get (71)
∑
ϕ
∈
T
a
A
k
,
n
-
k
ω
A
ϕ
=
∑
j
=
k
n
n
j
q
-
n
-
k
a
2
-
a
2
q
n
-
j
∑
φ
∈
B
-
j
-
k
∏
c
∈
φ
ω
∗
c
,where B
-
j
-
k is the set of all tableaux φ with j
-
k columns of weights ω
∗
(
j
i
)
=
[
j
i
-
a
1
]
q for each i
=
1,2
,
…
,
j
-
k. Clearly B
-
j
-
k
=
T
a
1
A
(
k
,
j
-
k
). Hence, (72)
∑
ϕ
∈
T
a
A
k
,
n
-
k
ω
A
ϕ
=
∑
j
=
k
n
n
j
q
-
n
-
k
a
2
-
a
2
q
n
-
j
∑
φ
∈
T
a
1
A
k
,
j
-
k
ω
A
φ
.Applying Theorem 13, we obtain the following theorem.Theorem 14.
Theq-analogue S
a
∗
[
n
,
k
]
q satisfies the following identity: (73)
S
a
∗
n
,
k
q
=
∑
j
=
k
n
-
1
n
-
j
n
j
q
-
n
-
k
a
2
a
2
q
n
-
j
S
a
1
∗
j
,
k
q
,where a
=
a
1
+
a
2 for some numbers a
1 and a
2.Supposeϕ
1
is a tableau withk
-
m columns whose lengths are in the set {
0,1
,
…
,
m
},
ϕ
2
is a tableau withn
-
k
-
j columns whose lengths are in the set {
m
+
1
,
m
+
2
,
…
,
m
+
j
+
1
}.Then, (74)
ϕ
1
∈
T
A
1
m
,
k
-
m
,
ϕ
2
∈
T
A
2
j
,
n
-
k
-
j
,where A
1
=
{
0,1
,
…
,
m
} and A
2
=
{
m
+
1
,
m
+
2
,
…
,
m
+
j
+
1
}. Notice that, by joining the columns of ϕ
1 and ϕ
2, we obtain an A-tableau ϕ with n
-
m
-
j columns whose lengths are in the set A
=
A
1
∪
A
2
=
{
0,1
,
…
,
m
+
j
+
1
}. That is, ϕ
∈
T
A
(
m
+
j
+
1
,
n
-
m
-
j
). Then,(75)
∑
ϕ
∈
T
A
m
+
j
+
1
,
n
-
m
-
j
ω
A
ϕ
=
∑
k
=
m
n
-
j
∑
ϕ
1
∈
T
A
1
m
,
k
-
m
ω
A
1
ϕ
1
∑
ϕ
2
∈
T
A
2
j
,
n
-
k
-
j
ω
A
2
ϕ
2
.Note that(76)
∑
ϕ
2
∈
T
A
2
j
,
n
-
k
-
j
ω
A
2
ϕ
2
=
∑
ϕ
2
∈
T
A
2
j
,
n
-
k
-
j
∏
c
∈
ϕ
2
c
-
a
q
=
∑
m
+
1
≤
g
1
≤
⋯
≤
g
n
-
k
-
j
≤
m
+
j
+
1
∏
i
=
1
n
-
k
-
j
g
i
-
a
q
=
∑
0
≤
g
1
≤
⋯
≤
g
n
-
k
-
j
≤
j
∏
i
=
1
n
-
k
-
j
g
i
+
m
+
1
-
a
q
.Thus,(77)
∑
0
≤
g
1
≤
⋯
≤
g
n
-
m
-
j
≤
m
+
j
+
1
∏
i
=
1
n
-
m
-
j
g
i
-
a
q
=
∑
k
=
m
n
-
j
∑
0
≤
g
1
≤
⋯
≤
g
k
-
m
≤
m
∏
i
=
1
k
-
m
g
i
-
a
q
∑
0
≤
g
1
≤
⋯
≤
g
n
-
k
-
j
≤
j
∏
i
=
1
n
-
k
-
j
g
i
+
m
+
1
-
a
q
.By (53), we obtain the following theorem.Theorem 15.
Theq-analogue S
a
∗
[
n
,
k
]
q satisfies the following convolution-type identity:(78)
S
a
∗
n
+
1
,
m
+
j
+
1
q
=
∑
k
=
0
n
S
a
∗
k
,
m
q
S
a
-
m
-
1
∗
n
-
k
,
j
q
.The next theorem provides another form of convolution-type identity.Theorem 16.
Theq-analogue S
a
∗
[
n
,
k
]
q satisfies the following second form of convolution formula: (79)
S
a
∗
m
+
j
,
n
q
=
∑
k
=
m
n
-
j
S
a
∗
m
,
k
q
S
a
-
k
∗
j
,
n
-
k
q
.Proof.
Letϕ
1
be a tableau withm
-
k columns whose lengths are in A
1
=
{
0,1
,
…
,
k
},
ϕ
2
be a tableau withj
-
n
+
k columns whose lengths are in A
2
=
{
k
,
k
+
1
,
…
,
n
}.Then, ϕ
1
∈
T
A
1
(
k
,
m
-
k
) and ϕ
2
∈
T
A
2
(
n
-
k
,
j
-
n
+
k
). Using the same argument above, we can easily obtain the convolution formula.


## 3.1. Combinatorics ofA-Tableaux

Suppose that, for some numbersa
1 and a
2, we have a
=
a
1
+
a
2. Then, (53) yields (56)
S
a
∗
n
,
k
q
=
∑
0
≤
j
1
≤
j
2
≤
⋯
≤
j
n
-
k
≤
k
∏
i
=
1
n
-
k
j
i
-
a
1
-
a
2
q
.That is, for any ϕ
∈
T
a
A
(
k
,
n
-
k
), (57)
ω
A
ϕ
=
∏
c
∈
ϕ
j
i
-
a
1
-
a
2
q
,where |
c
|
∈
{
0,1
,
2
,
…
,
k
}. Note that the weight of each column of ϕ can be considered as a finite sum with additive constant a
2; that is, for each c
∈
ϕ, we can write(58)
ω
c
=
1
q
a
2
ω
∗
c
-
a
2
q
,where ω
∗
(
|
c
|
)
=
[
|
c
|
-
a
1
]
q. The following theorem determines how an additive constant affects the recurrence formula for S
a
[
n
,
k
]
q.From Theorem13,(59)
S
a
∗
n
,
k
q
=
∑
ϕ
∈
T
a
A
k
,
n
-
k
ω
A
ϕ
=
∑
ϕ
∈
T
a
A
k
,
n
-
k
∏
c
∈
ϕ
ω
c
,where (60)
ω
A
ϕ
=
∏
c
∈
ϕ
c
-
a
q
,
where
c
∈
0,1
,
…
,
k
=
∏
i
=
1
n
-
k
j
i
-
a
q
,
where
j
i
∈
0,1
,
…
,
k
.If a
=
a
1
+
a
2 for some a
1 and a
2, then, by (58), (61)
ω
A
ϕ
=
∏
i
=
1
n
-
k
1
q
a
2
ω
∗
j
i
-
a
2
q
,
where
ω
∗
j
i
=
j
i
-
a
1
q
=
q
-
n
-
k
a
2
ω
∗
j
1
-
a
2
q
ω
∗
j
2
-
a
2
q
⋯
ω
∗
j
n
-
k
-
a
2
q
=
q
-
n
-
k
a
2
∑
l
=
0
n
-
k
-
a
2
q
n
-
k
-
l
∑
j
1
≤
q
1
≤
q
2
≤
⋯
≤
q
l
≤
j
n
-
k
∏
i
=
1
l
ω
∗
q
i
.SupposeB
ϕ is the set of all A-tableaux corresponding to ϕ such that, for each ψ
∈
B
ϕ, either(62)
ψ
has  no  column  whose  weight  is
-
a
2
q
,
or
ψ
has  one  column  whose  weight  is
-
a
2
q
,
or
ψ
has  two  columns  whose  weights  are
-
a
2
q
,
or
⋮
ψ
has
n
-
k
columns  whose  weights  are
-
a
2
q
.Then, we may write (63)
ω
A
ϕ
=
∑
ψ
∈
B
ϕ
ω
A
ψ
.Now, if r columns in ψ have weights other than -
[
a
2
]
q, then(64)
ω
A
ψ
=
∏
c
∈
ψ
ω
∗
c
=
q
-
n
-
k
a
2
-
a
2
q
n
-
k
-
r
∏
i
=
1
r
ω
∗
q
i
,where q
1
,
q
2
,
…
,
q
r
∈
{
j
1
,
j
2
,
…
,
j
n
-
k
}. Note that, for each r, there correspond (65)
n
-
k
rtableaux with r columns having weights ω
∗
(
j
i
)
=
[
j
i
-
a
1
]
q. It can easily be verified that (66)
T
a
A
k
,
n
-
k
=
n
-
k
+
k
n
-
k
=
n
n
-
k
=
n
k
.Thus, ∀
ϕ
∈
T
a
A
(
k
,
n
-
k
), B
ϕ contains a total of (67)
n
k
n
-
k
rtableaux with r columns of weights ω
∗
(
j
i
). However, only (68)
r
+
k
rtableaux with r columns in B
ϕ are distinct. Hence, every distinct tableau ψ with r columns of weights other than -
[
a
2
]
q appears(69)
n
k
n
-
k
r
r
+
k
r
=
n
r
+
ktimes in the collection. Thus, (70)
∑
ϕ
∈
T
a
A
k
,
n
-
k
ω
A
ϕ
=
∑
r
=
0
n
-
k
n
r
+
k
q
-
n
-
k
a
2
-
a
2
q
n
-
k
-
r
∑
φ
∈
B
-
r
∏
c
∈
φ
ω
∗
c
,where B
-
r denotes the set of all tableaux φ having r columns of weights ω
∗
(
j
i
)
=
[
j
i
-
a
1
]
q. Reindexing the double sum, we get (71)
∑
ϕ
∈
T
a
A
k
,
n
-
k
ω
A
ϕ
=
∑
j
=
k
n
n
j
q
-
n
-
k
a
2
-
a
2
q
n
-
j
∑
φ
∈
B
-
j
-
k
∏
c
∈
φ
ω
∗
c
,where B
-
j
-
k is the set of all tableaux φ with j
-
k columns of weights ω
∗
(
j
i
)
=
[
j
i
-
a
1
]
q for each i
=
1,2
,
…
,
j
-
k. Clearly B
-
j
-
k
=
T
a
1
A
(
k
,
j
-
k
). Hence, (72)
∑
ϕ
∈
T
a
A
k
,
n
-
k
ω
A
ϕ
=
∑
j
=
k
n
n
j
q
-
n
-
k
a
2
-
a
2
q
n
-
j
∑
φ
∈
T
a
1
A
k
,
j
-
k
ω
A
φ
.Applying Theorem 13, we obtain the following theorem.Theorem 14.
Theq-analogue S
a
∗
[
n
,
k
]
q satisfies the following identity: (73)
S
a
∗
n
,
k
q
=
∑
j
=
k
n
-
1
n
-
j
n
j
q
-
n
-
k
a
2
a
2
q
n
-
j
S
a
1
∗
j
,
k
q
,where a
=
a
1
+
a
2 for some numbers a
1 and a
2.Supposeϕ
1
is a tableau withk
-
m columns whose lengths are in the set {
0,1
,
…
,
m
},
ϕ
2
is a tableau withn
-
k
-
j columns whose lengths are in the set {
m
+
1
,
m
+
2
,
…
,
m
+
j
+
1
}.Then, (74)
ϕ
1
∈
T
A
1
m
,
k
-
m
,
ϕ
2
∈
T
A
2
j
,
n
-
k
-
j
,where A
1
=
{
0,1
,
…
,
m
} and A
2
=
{
m
+
1
,
m
+
2
,
…
,
m
+
j
+
1
}. Notice that, by joining the columns of ϕ
1 and ϕ
2, we obtain an A-tableau ϕ with n
-
m
-
j columns whose lengths are in the set A
=
A
1
∪
A
2
=
{
0,1
,
…
,
m
+
j
+
1
}. That is, ϕ
∈
T
A
(
m
+
j
+
1
,
n
-
m
-
j
). Then,(75)
∑
ϕ
∈
T
A
m
+
j
+
1
,
n
-
m
-
j
ω
A
ϕ
=
∑
k
=
m
n
-
j
∑
ϕ
1
∈
T
A
1
m
,
k
-
m
ω
A
1
ϕ
1
∑
ϕ
2
∈
T
A
2
j
,
n
-
k
-
j
ω
A
2
ϕ
2
.Note that(76)
∑
ϕ
2
∈
T
A
2
j
,
n
-
k
-
j
ω
A
2
ϕ
2
=
∑
ϕ
2
∈
T
A
2
j
,
n
-
k
-
j
∏
c
∈
ϕ
2
c
-
a
q
=
∑
m
+
1
≤
g
1
≤
⋯
≤
g
n
-
k
-
j
≤
m
+
j
+
1
∏
i
=
1
n
-
k
-
j
g
i
-
a
q
=
∑
0
≤
g
1
≤
⋯
≤
g
n
-
k
-
j
≤
j
∏
i
=
1
n
-
k
-
j
g
i
+
m
+
1
-
a
q
.Thus,(77)
∑
0
≤
g
1
≤
⋯
≤
g
n
-
m
-
j
≤
m
+
j
+
1
∏
i
=
1
n
-
m
-
j
g
i
-
a
q
=
∑
k
=
m
n
-
j
∑
0
≤
g
1
≤
⋯
≤
g
k
-
m
≤
m
∏
i
=
1
k
-
m
g
i
-
a
q
∑
0
≤
g
1
≤
⋯
≤
g
n
-
k
-
j
≤
j
∏
i
=
1
n
-
k
-
j
g
i
+
m
+
1
-
a
q
.By (53), we obtain the following theorem.Theorem 15.
Theq-analogue S
a
∗
[
n
,
k
]
q satisfies the following convolution-type identity:(78)
S
a
∗
n
+
1
,
m
+
j
+
1
q
=
∑
k
=
0
n
S
a
∗
k
,
m
q
S
a
-
m
-
1
∗
n
-
k
,
j
q
.The next theorem provides another form of convolution-type identity.Theorem 16.
Theq-analogue S
a
∗
[
n
,
k
]
q satisfies the following second form of convolution formula: (79)
S
a
∗
m
+
j
,
n
q
=
∑
k
=
m
n
-
j
S
a
∗
m
,
k
q
S
a
-
k
∗
j
,
n
-
k
q
.Proof.
Letϕ
1
be a tableau withm
-
k columns whose lengths are in A
1
=
{
0,1
,
…
,
k
},
ϕ
2
be a tableau withj
-
n
+
k columns whose lengths are in A
2
=
{
k
,
k
+
1
,
…
,
n
}.Then, ϕ
1
∈
T
A
1
(
k
,
m
-
k
) and ϕ
2
∈
T
A
2
(
n
-
k
,
j
-
n
+
k
). Using the same argument above, we can easily obtain the convolution formula.


## 4.q-Noncentral Bell Number and Its Hankel Transform

In this section, we define aq-analogue of the noncentral Bell numbers and obtain some combinatorial properties that will be used to establish its Hankel transform.Definition 17.
Aq-analogue of the noncentral Bell numbers, denoted by B
~
n
,
a
q, is defined by (80)
B
~
n
,
a
q
=
∑
k
=
0
n
S
~
a
n
,
k
,where S
~
a
[
n
,
k
]
=
q
-
k
a
S
a
∗
[
n
,
k
]
q
=
q
-
k
2
S
a
[
n
,
k
]
q. For brevity, we use the term q
-noncentral Bell numbers for B
~
n
,
a
q.We observe that, multiplying both of (22) by q
k
a
-
k
2, we obtain(81)
q
k
a
-
k
2
S
a
n
,
k
q
=
q
k
-
1
a
-
k
-
1
2
S
a
n
-
1
,
k
-
1
q
+
k
-
a
q
q
k
a
-
k
2
S
a
n
-
1
,
k
q
,
S
a
∗
n
,
k
q
=
S
a
∗
n
-
1
,
k
-
1
q
+
k
-
a
q
S
a
∗
n
-
1
,
k
q
,
S
~
a
n
,
k
q
=
q
-
a
S
~
a
n
-
1
,
k
-
1
q
+
k
-
a
q
S
~
a
n
-
1
,
k
q
.Note that when a
=
0, we get(82)
S
~
0
n
,
k
q
=
S
~
0
n
-
1
,
k
-
1
q
+
k
q
S
~
0
n
-
1
,
k
q
,which coincides with the recurrence relation of the modified q-Stirling numbers of the second kind S
~
[
n
,
k
] in [11] and that of the q-Stirling numbers of the second kind in [15]. In fact,(83)
S
~
0
n
,
k
q
=
q
-
k
2
S
0
n
,
k
q
=
q
-
k
2
S
n
,
k
q
=
S
~
n
,
k
.That is, S
~
a
[
n
,
k
]
q is a proper q-analogue of S
a
(
n
,
k
) that generalizes S
~
[
n
,
k
].Note that our objective is to define aq-analogue of B
n
,
a such that when a
=
0, it reduces to the q-exponential number (84)
e
~
n
=
e
~
n
1
=
∑
k
=
0
n
S
~
n
,
k
=
φ
n
1
,in [11, 15], whose Hankel transform is known to be (85)
H
e
~
n
=
H
φ
n
1
=
q
n
+
1
3
∏
j
=
0
n
j
!(see [11, 15, 16]). Hence, it is appropriate to define a q-analogue of the noncentral Bell numbers through S
~
a
[
n
,
k
]
q.Clearly, whenq
→
1, B
~
n
,
a
q
→
B
n
,
a, the noncentral Bell numbers. The following theorem contains certain recurrence relations for B
~
n
,
a
q.Theorem 18.
Theq-noncentral Bell numbers satisfy the following relation: (86)
q
n
B
~
n
,
a
q
=
∑
j
=
0
n
-
1
n
-
j
n
j
B
~
j
,
a
-
1
q
.Proof.
By making use of Theorem14, with a
1
=
a
-
1 and a
2
=
1, we have (87)
q
-
k
a
S
a
∗
n
,
k
q
=
q
-
k
a
∑
j
=
k
n
-
1
n
-
j
n
j
q
-
n
-
k
S
a
-
1
∗
j
,
k
q
,
(88)
∑
k
=
0
n
q
-
k
a
S
a
∗
n
,
k
q
=
∑
k
=
0
n
∑
j
=
k
n
-
1
n
-
j
n
j
q
-
n
q
-
k
a
-
1
S
a
-
1
∗
j
,
k
q
=
∑
j
=
0
n
-
1
n
-
j
n
j
q
-
n
∑
k
=
0
j
q
-
k
a
-
1
S
a
-
1
∗
j
,
k
q
.Then,(89)
B
~
n
,
a
q
=
∑
j
=
0
n
-
1
n
-
j
n
j
q
-
n
B
~
j
,
a
-
1
q
,which is exactly the desired relation.The following corollary is a direct consequence of Theorem18 which can be proved using the inversion formula by Riordan [2, 3].Corollary 19.
Theq-noncentral Bell numbers satisfy the following relation: (90)
B
~
n
,
a
-
1
q
=
∑
j
=
0
n
n
j
q
j
B
~
j
,
a
q
.To establish the Hankel transform ofB
~
n
,
a
q, we need the concept of rising k-binomial transform by Spivey and Steil [17] as well as its property in relation to Hankel transform.Definition 20 (Spivey-Steil [17]).
Therising
k
-binomial transform
R of a sequence A
=
{
a
n
} is the sequence R
(
A
;
k
)
=
{
r
n
}, where r
n is given by (91)
r
n
=
∑
j
=
0
n
n
j
k
j
a
j
,
k
≠
0
.We useR
(
A
,
k
) to denote the set of rising k-binomial transforms of A. That is, R
(
A
,
k
)
=
{
r
n
}. Then, we have the following theorem by Spivey and Steil.Theorem 21 (Spivey-Steil [17]).
Given a sequenceA
=
{
a
0
,
a
1
,
…
}, let H
(
A
)
=
{
h
n
}. Then,(92)
H
R
A
,
0
=
a
0
,
0,0
,
…
.If k
≠
0, (93)
H
R
A
,
k
=
k
n
n
+
1
h
n
.Now, we are ready to state the main result of the paper.Theorem 22.
Fora
=
0,1
,
2
,
…, the Hankel transform of the sequence of q-noncentral Bell numbers {
B
~
n
,
a
q
} is given by (94)
H
B
~
n
,
a
q
=
q
-
a
n
n
+
1
+
n
+
1
3
∏
j
=
0
n
j
!
.Proof.
We begin with the relation thatB
~
n
,
0
q
=
e
~
n. Hence, (95)
H
B
~
n
,
0
q
=
H
e
~
n
=
q
n
+
1
3
∏
j
=
0
n
j
!
.From Corollary 19, we see that B
~
n
,
a
-
1
q is the binomial transform of q
n
B
~
n
,
a
q. This means that (96)
B
q
n
B
~
n
,
a
q
=
B
~
n
,
a
-
1
q
.Hence, by Layman’s Theorem [1],(97)
H
B
q
n
B
~
n
,
a
q
=
H
q
n
B
~
n
,
a
q
.That is,(98)
H
B
~
n
,
a
-
1
q
=
H
q
n
B
~
n
,
a
q
.Now, Corollary 19 can also be stated as B
~
n
,
a
-
1
q is the rising q-binomial transform of B
~
n
,
a
q. Using Spivey-Steil Theorem, with A
=
{
B
~
n
,
a
q
}, h
n
=
H
(
B
~
n
,
a
q
), and r
n
=
B
~
n
,
a
-
1
q, we have(99)
H
B
~
n
,
a
-
1
q
=
q
n
n
+
1
H
B
~
n
,
a
q
.We observe that when a
=
1 and using (95), we have (100)
H
B
~
n
,
1
q
=
q
-
n
n
+
1
H
B
~
n
,
0
q
=
q
-
n
n
+
1
+
n
+
1
3
∏
j
=
0
n
j
!
.Also, when a
=
2, (101)
H
B
~
n
,
2
q
=
q
-
n
n
+
1
H
B
~
n
,
1
q
=
q
-
n
n
+
1
q
-
n
n
+
1
+
n
+
1
3
∏
j
=
0
n
j
!
=
q
-
2
n
n
+
1
+
n
+
1
3
∏
j
=
0
n
j
!
.Continuing this argument, we obtain(102)
H
B
~
n
,
a
q
=
q
-
a
n
n
+
1
+
n
+
1
3
∏
j
=
0
n
j
!
.Remark 23.
Whena
=
0, the Hankel transform in (94) reduces to the Hankel transform of q-exponential numbers in (85) and that in [15]. By taking q
→
1, we further get (103)
H
B
n
=
∏
j
=
0
n
j
!
,the Hankel transform of the ordinary Bell numbers. The same Hankel transform is obtained when a
=
-
r and taking q
→
1, which is exactly the same result established by Mező [5] for the Hankel transform of r-Bell numbers B
r
(
n
). That is, (104)
H
B
r
n
=
H
B
n
,
-
r
=
∏
j
=
0
n
j
!
.Remark 24.
We can define another form ofq-analogue of the noncentral Bell numbers B
n
,
a as follows: (105)
B
n
,
a
q
=
∑
k
=
0
n
S
a
n
,
k
q
=
∑
k
=
0
n
q
k
2
-
r
a
S
a
∗
n
,
k
q
.In the paper by Cigler [18], certain generalization of q-Stirling numbers of the second kind, denoted by S
(
n
,
k
,
r
), was defined by means of the following recurrence relation: (106)
S
n
,
k
,
r
=
S
n
-
1
,
k
-
1
,
r
+
k
+
r
q
S
n
-
1
,
k
,
rwhich is equivalent to S
a
∗
[
n
,
k
]
q. That is, (107)
S
a
∗
n
,
k
q
=
S
n
,
k
,
-
a
.Moreover, the first form of generalized q-exponential polynomials in [18] was defined by (108)
Φ
n
x
,
r
=
∑
k
=
0
n
S
n
,
k
,
r
q
k
2
+
k
r
x
kwhose Hankel transform is given by (109)
H
Φ
n
x
,
r
=
q
2
n
+
1
3
+
2
r
n
+
1
2
x
n
+
1
2
∏
k
=
0
n
k
q
!
1
-
q
x
;
q
k
,where (
x
;
q
)
k
=
∏
j
=
0
k
-
1
‍
(
1
-
q
j
x
). With x
=
1 and r
=
-
a, we obtain(110)
Φ
n
1
,
-
a
=
B
n
,
a
q
.Then, by (109), (111)
H
B
n
,
a
q
=
q
2
n
+
1
3
-
2
a
n
+
1
2
∏
k
=
0
n
k
q
!
1
-
q
;
q
k
.Remark 25.
We can also define aq-analogue of the noncentral Bell numbers B
n
,
a
∗ as follows: (112)
B
^
n
,
a
q
=
∑
k
=
0
n
S
a
∗
n
,
k
q
.Another form of generalized q-exponential polynomials in [18] was defined by (113)
φ
n
x
,
r
=
∑
k
=
0
n
S
n
,
k
,
r
x
kwhose Hankel transform is given by (114)
H
φ
n
x
,
r
=
q
n
+
1
3
q
r
x
n
+
1
2
∏
k
=
0
n
k
q
!
.With x
=
1 and r
=
-
a, we obtain (115)
φ
n
1
,
-
a
=
B
^
n
,
a
q
.Then, by (114),(116)
H
B
^
n
,
a
q
=
q
n
+
1
3
-
a
n
+
1
2
∏
k
=
0
n
k
q
!
.



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*Source: 417327-2015-10-28.xml*