Hankel Transform of ( q , r )-Dowling Numbers

In this paper, the authors establish certain combinatorial interpretation for q-analogue of r-Whitney numbers of the second kind defined by Corcino and Cañete in the context of Atableaux. They derive convolution-type identities by making use of the combinatorics of Atableaux. Finally, they define a q-analogue of r-Dowling numbers and obtain some necessary properties including its Hankel transform.


Introduction
The binomial transform B of a sequence A = {a n } is the sequence {b n } defined by That is, B(A) = b n .It is one of the common and useful transforms that frequently appeared in the literature of integer sequences (see [16]).The inverse binomial transform (or inverse transform) C of a sequence A is the sequence {c n } defined by That is, C(A) = c n .
The Hankel 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 .The Hankel determinant h n of order of n of A is the determinant of the corresponding Hankel matrix of order n.That is, h n = det(H n ).The Hankel transform of the sequence A, denoted by H(A), is the sequence {h n } of Hankel determinants of A.
where F n is the nth Fibonacci numbers [12].One remarkable property of Hankel transform is established by Layman [12], which states that the Hankel transform of an integer sequence is invariant under binomial and inverse transforms.That is, if A is an integer sequence, B is binomial transform of A and C is the inverse transform of A, then This property played an important role in proving that the Hankel transform of the sequence of Bell number {B n )} [1] and that of r-Bell numbers {B n,r } [14] are equal.Recently, in the paper by R. Corcino and C. Corcino [7], 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 (see [5,8]), which are also known as (r, β)-Bell numbers.In the same paper, the authors have made an attempt to establish the Hankel transform for the q-analogue of (r, β)-Bell numbers.However, they are not successful with their attempt and have conjectured that the Hankel transform for the q-analogue of (r, β)-Bell numbers when r = 0 is equal to for some number f (n, k), which is a function of n and k.With this, the present authors have decided to use other method.Recently, R. Corcino et al. [9] have successfully established the Hankel transform for the q-analogue of noncentral Bell numbers.This motivates the present authors to use this method to establish the Hankel transform for the q-analogue of (r, β)-Bell numbers G n,r,β .It is important to note that the numbers G n,r,β are equivalent to the r-Dowling numbers D m,r (n), which are defined as the sum of r-Whitney numbers of the second kind, denoted by W m,r (n, k).That is, The term "r-Dowling numbers" was introduced by Cheon and Jung [3].
2. A q-Analogue of W m,r (n, k): Second Form A q-analogue of both kinds of Stirling numbers was first defined by Carlitz in [2].The second kind of which, known as q-Stirling numbers of the second kind, is defined in terms of the following recurrence relation in 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) A different way of defining q-analogue of Stirling numbers of the second kind has been adapted in the paper by [10] which is given as follows This type of q-analogue gives the Hankel transform of q-exponential polynomials and numbers which are certain q-analogue of Bell polynomials and numbers.Recently, a qanalogue of r-Whitney numbers of the second kind was defined by Corcino and Cañete [6] parallel to the definition for q-analogue of noncentral Stirling numbers of the second kind as follows: Definition 1.For non-negative integers n and k, and real number a, a q-analogue W where Remark 1.When m = 1 and r = 0, the relation (5) reduces to (4).This implies that The q-analogue W m,r [n, k] q satisfies the following properties: Vertical and Horizontal Recurrence Relations Horizontal Generating Function Explicit Formula Exponential Generating Function Rational Generating Function We now define another form of q-analogue of r-Whitney numbers of the second, denoted by W * m,r [n, k] q , as follows Hence, All other properties parallel to those of W m,r [n, k] q can easily be established by imbedding the factor q −kr−m( k 2 ) in the derivations or multiply directly to the resulting identities/formula.Definition 2. [13] An A-tableau is a list φ of column c of a Ferrer's diagram of a partition λ(by decreasing order of length) such that the lengths |c| are part of the sequence A = (r i ) i≥0 , a strictly increasing sequence of nonnegative integers.
Let ω be a function from the set of nonnegative integers N to a ring K. Suppose Φ is an A-tableau with l columns of lengths |c| ≤ h.We use T A r (h, l) to denote the set of such A-tableaux.Then, we set 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 an A-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 1.Let ω : N → K denote a function from N to a ring K (column weights according to length) which is defined by ω(|c|) = [m|c| + r] q where r is a complex number, and |c| is the length of column l of an A-tableau in Suppose that for some numbers r 1 and r 2 , we have r = r 1 + r 2 .Then, equation ( 13) yields That is, for any 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 r 2 , that is, for each c ∈ φ, we can write where ω * (|c|) = [m|c| + r 1 ] q .The following theorem determines how an additive constant affects the recurrence formula for W m,r [n, k] q .From Theorem 1, where [mj i + r] q , where j i ∈ {0, 1, . . ., k}.
If r = r 1 + r 2 for some r 1 and r 2 , then by (14), Suppose B φ is the set of all A-tableaux corresponding to φ such that for each ψ ∈ B φ , either ψ has no column whose weight is [r 2 ] q , or ψ has one column whose weight is [r 2 ] q , or ψ has two columns whose weights are [r 2 ] q , or . . .
Then, we may write Now, if l columns in ψ have weights other than [r 2 ] q , then where q 1 , q 2 , . . ., q r ∈ {j 1 , j 2 , . . ., j n−k }.Note that for each l, there corresponds n − k l tableaux with l columns having weights ω * (j i ) = [mj i + r 1 ] q .It can be easily verified that, tableaux with l columns of weights ω * (j i ).However, only l+k l tableaux with l columns in B φ are distinct.Hence, every distinct tableaux ψ with l columns of weights other than where Bl denotes the set of all tableaux ϕ having l columns of weights ω * (j i ) = [mj i +r 1 ] q .
Reindexing the double sum, we get where Bj−k is the set of all tableaux ϕ with j − k columns of weights ω * (j Applying Theorem 1, we obtain the following theorem.
Theorem 2. The q-analogue W * m,r [n, k] q satisfies the following identity where r = r 1 + r 2 for some numbers r 1 and r 2 .
Theorem 3. The q-analogue W * m,r [n, k] satisfies the following convolution-type identity The next theorem provides another form of convolution-type identity.
Theorem 4. The q-analogue W * m,r [n, k] q satisfies the following second form of convolution formula Proof.Let φ 1 be a tableau with s − k columns whose lengths are in A 1 = {0, 1, . . ., k}, and φ 2 be a tableau with j − n + k columns whose lengths are in Using the same argument above, we can easily obtain the convolution formula.

(q, r)−Dowling Number and Its Hankel Transform
In this section, we define a q-analogue of the r-Dowling numbers and obtain some combinatorial properties that will be used to establish its Hankel transform.
A q-analogue of the r-Dowling numbers, denoted by D m,r [n] q , is defined by For brevity, we use the term (q, r)-Dowling numbers for D m,r [n] q .
For instance, the Hankel transform of the sequence of Catalan numbers C = { 1 n+1 2n n } ∞ n=1 , is given by 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 the nth Catalan numbers, has the Hankel transform [n]!(z) ( n+1 2 ) .It can easily be verified that the Hankel transform ofēq,n [z] = n k=0 S q [n, k]z n−k (18)