1. Introduction

The theory of Hankel matrices has been previously studied by some mathematician and its connections in some areas of mathematics, physics and computer science (see, the works of Desainte-Catherine and Viennot ^[9], Garcia-Armas and Sethuraman ^[11], Tamm ^[22], Vein and Dale ^[23]). Though, Hankel determinants had been previously studied (see, for example, Aigner ^[1], Radoux ^[19], Ehrenborg ^[10]), the term Hankel Transform was first introduced in Sloane’s sequence A055878 ^[20] and first studied by Layman ^[15]. Layman used the notion of binomial transform (b_n) of a sequence (a_n) given by

and the invert transform

in establishing some properties of the Hankel transform including the theorem which states that any integer sequence has the same Hankel transform as its binomial or invert transform.

Layman found out that some sequences have the same Hankel transform. For instance, the sequence of Catalan numbers {1, 1, 2, 3, 14, 42, ...} (sequence A000108 in the EIS) and approximately twenty sequences have the same Hankel transform {1, 1, 1, ...}. The following are some of the sequences with the same Hankel trans- form.

Also, Layman and Michael Somos found ten sequences (A055209) in the EIS whose Hankel transform is

which was shown theoretically by Radoux ^[19] to be the Hankel transform of the derangements, or rencontres numbers (A000166).

Several later studies of Hankel transform of some integer sequences were estab- lished. Among them were:

1. Cvetcoć et al. ^[8], who established the Hankel transform of the sequence of the sum of two adjacent Catalan numbers. More precisely, if we let a_n = C_n+C_n+1 where C_n is the nth Catalan number, then the Hankel transform of an is

where Fn is nth Fibonnaci number.

2. Armas and Sethuraman ^[11], who established the Hankel transform of central binomial coeffficients which is stated as follows:

The zeroth Hankel transform of the sequence is the sequence , and the first Hankel transform is the sequence ,

3. Spivey and Steil ^[21], French (2007), Chamberland and French (2007), Rajković, Ivković and Barry (2007), who established the k-binomial transform and Han- kel transform, transformation in preserving the Hankel transform, generalized Catalan numbers and Hankel transformations, and the Hankel transform of the sum of consecutive generalized Catalan numbers, respectively.

4. Aigner ^[1], who established a characterization of the sequence of Bell numbers (B_n) and proved that this sequence has the Hankel transform which is given by

5. Mezo ^[16], who recently proved that the Bell numbers and r-Bell numbers ^[18] have the same Hankel transform.

In this present study, certain generalization of Bell numbers which is defined as the sum of noncentral Stirling numbers of the second by M. Koutras ^[14], will be established. It will also be shown that these generalized Bell numbers has the same Hankel transform as that of the sequence of ordinary Bell numbers.

2. The Noncentral Bell Numbers

In 1982, M. Koutras ^[14] introduced the noncentral Stirling numbers of first and second kind. These numbers denoted by s_a(n, k) and S_a(n, k) are defined as the coefficients of the following expansions, with parameter a,

(1)

(2)

where and

The following theorems contain some combinatorial identities of the noncentral Stirling numbers of both kind which are established by Koutras ^[14].

Theorem 2.1. The noncentral Stirling numbers of the first and second kind satisfy the recurrence relations

(3)

(4)

where and if or

Note that if a = 0 in (3) and (4) , then we have

Thus, the ordinary Stirling numbers can be expressed as

Theorem 2.2. The numbers and have the following exponential generating functions

(5)

(6)

Theorem 2.3. The numbers and Sa(n; k) have the following explicit formula

(7)

(8)

Now, let us define the noncentral Bell numbers parallel to the definition of the ordinary Bell numbers.

Definition 2.4. The noncentral Bell numbers, denoted by B_a(n), are defined by

In particular, B₀(n) = B(n), the ordinary Bell numbers.

Using the exponential generating function of the noncentral Stirling numbers of the second kind, we can obtain the following exponential generating function for B_a(n).

Theorem 2.5. The noncentral Bell numbers have the following generating functions

(9)

Proof. By making use of the exponential generating function (6) of we have,

Hence, the exponential generating function of is

If a = 0, (9) becomes

the exponential generating function of the ordinary Bell numbers.

The next theorem contains a kind of Dobinski formula for

Theorem 2.6. The noncentral Bell numbers can be written in the form of a convergent series

Proof. Applying the exponential generating function of in (9),

Comparing the coefficient of , we obtain

The following theorem contains some relations which are useful in establishing the alternative proof of the claim that the sequence of has the same Hankel transform as that of the sequence of This is a kind of a recurrence relation of

Theorem 2.7. The noncentral Bell numbers satisfy the relations

(10)

(11)

Proof. Multiplying to both side of (9) with a being replaced with , we have

Comparing the coefficient of , we obtain the following relation

Similarly, multiplying to both side of (9), we have

Comparing the coefficient of

Remark 2.8. Theorem 4.1.4 implies that is the binomial transform of or is the inverse transform of .

3. The Hankel Transform of Noncentral Bell Numbers

Let be the in_nite lower triangular matrix defined recursively by,

(12)

with the intial condition

The following lemma contains the exponential generating function of the kth column entries of .

Lemma 3.1. Let be the exponential generating function of the kth column of

then

where That is, the 0-column entries of are .

Proof. By making use of the recurrence relation in (12) we obtain

(13)

With the left-hand side (LHS) of (13) yields

On the other hand, the right-hand side (RHS) of (13) gives

This implies that the generating function is a unique solution to the differential equation (13). Thus, the exponential generating function of the kth column of is given by

Lemma 3.2. Let be the nth row of . Define then for all n and l.

Proof. We prove this by induction on If we have Since Suppose that Suppose that and all Then by (12) and intechanging the summation we have,

By the inductive hypothesis,

We are now ready to introduce the following theorem.

Theorem 3.3. The noncentral Bell numbers have the Hankel transform

Proof. Let be the submatrix of consisting of the rows and columns numbered 0 to n. Clearly, since is a lower triangular matrix with diagonal 1. It follows that Let Then

By Lemma 3.2,

where

That is,

Thus,

The theorem can also be proved without using Lemma 3.2. One can use the fact that the ordinary Bell numbers. This means that

That is, the Hankel transform of is By Theorem 2.7, is the binomial transform of and so, by Layman's theorem, and have the same Hankel transform. Again by Theorem 2.7, implies that is the binomial transform of So, by Layman's Theorem, and have the same Hankel transform. Continuing this process and again, since

by induction, and have the same Hankel transform. Hence, and have the same Hankel transform. Thus,

For possible future research, it is interesting to establish q-analogues of the noncentral Stirling and Bell numbers and to determine their Hankel transforms. It will be more interesting if one can establish connections with those q-analogues of Stirling and Bell numbers via normal ordering expressions of creation and annihilation operators (see ^{[12, 13]}).

4. Further Generalization

The r-Whitney numbers of the second kind ^[17], denoted by are certain extension of noncentral Stirling numbers. In particular, Properties of noncentral Stirling numbers of the second kind can be deduced from those of r-Whitney numbers of the second kind by taking and . One may see ^[17] for a more detailed discussion of r-Whitney numbers of the second kind.

The Dowling numbers, denoted by , were defined as the sum of Whitney numbers of the second kind (see ^{[2, 3]}). Hence, one may define the r-Dowling numbers, say denoted by as

(14)

These numbers are equivalent to (r, β)-Bell numbers ^[7] and they are certain extension of non-central Bell numbers. In fact,

On the other hand, the (r, β)-Bell numbers, denoted by , were shown to have the following Hankel transform ^[7]

(15)

This Hankel transform has been shown using the same method employed to obtain the above alternative solution for the Hankel transform of In this section, we are going to give an alternative proof for (15) following the method in doing the first proof for the Hankel transform of

Let be the infinite lower triangular matrix defined recursively by,

(16)

where if , and if Then we have the following lemma.

Lemma 4.1. Let be the exponential generating function of the kth column of matrix , that is,

Then

(17)

where and That is, the 0-column entries of are

Proof. Using the recurrence relation in (16), we obtain

This implies that

(18)

With , the left-hand side of (18) yields

While the right-hand side of (18) gives

which shows that the function where is a unique solution to the differential equation (18). Thus, the exponential generating function of the kth column of M is given by

Lemma 4.2. Let be the nth row of Define

Then for all n and m.

Proof. By induction, if n = 0 we have

Since

Suppose that holds for and all Then by (16)

By interchanging and reindexing the summation, we have

By (16),

By the inductive hypothesis,

Theorem 4.3. The (r, β)-Bell numbers have the Hankel Transform

where (Bell numbers) and (Non-central Bell numbers).

Proof. Let be the lower triangular submatrix of consisting of the rows and columns numbered 0 to n. Then is a matrix with diagonal 1. It follows that This implies that the determinant of upper triangular submatrix is one; that is, Let Then

By Theorem 4.2,

where

That is,

Thus,

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