The Translated Dowling Polynomials and Numbers

More properties for the translated Whitney numbers of the second kind such as horizontal generating function, explicit formula, and exponential generating function are proposed. Using the translated Whitney numbers of the second kind, we will define the translated Dowling polynomials and numbers. Basic properties such as exponential generating functions and explicit formula for the translated Dowling polynomials and numbers are obtained. Convexity, integral representation, and other interesting identities are also investigated and presented. We show that the properties obtained are generalizations of some of the known results involving the classical Bell polynomials and numbers. Lastly, we established the Hankel transform of the translated Dowling numbers.


Introduction
In 1996, the classical Whitney numbers of the second kind ( , ) of Dowling Lattices was introduced by Benoumhani [1].
( , ) satisfy the recurrence relation: Other fundamental properties of these numbers were already established by Benoumhani in [1,2]. The numbers ( , ) can be shown to be a kind of generalization of the famous Stirling numbers of the Second kind ( , ) when the parameter equals to 1. That is, Recently, a translated version of the Whitney numbers of the second kind was introduced by Belbachir and Bousbaa [3] which they named translated Whitney numbers of the second kind, denoted bỹ( ) ( , ).̃( ) ( , ) actually counts the number of partitions of a set with elements into subsets such that the elements of each subset can mutate in ways, except the dominant one. To compute the first few values of these numbers, the following recurrence relation was obtained in [3]: The classical Stirling numbers of the second kind can also be obtained from these numbers when = 1. On the otherhand, the classical Dowling numbers ( ) are defined to be the sum of ( , ). That is, and can be computed using the explicit formula: ( ) is known to be a generalization of the classical Bell numbers which is the sum of the Stirling numbers of the second kind ( , ). In this paper, we will define the translated Dowling numbers as the sum of̃( ) ( , ). The content of this paper is summarized as follows. In Section 2, we will introduce some basic properties for the numbers̃( ) ( , ). In Section 3, we will define the translated Dowling polynomials and numbers and derive some of their basic properties. In Section 4, we investigate convexity and integral representation of the translated Dowling polynomials and numbers. In Section 5, more properties of translated Dowling polynomials and numbers are presented, and in Section 6, we obtain the Hankel transform of the translated Dowling numbers.

Some Properties of̃( ) ( , )
Interesting properties of̃( ) ( , ) can also be obtained parallel to those done in [1]. For instance, by induction on , the following horizontal generating function can easily be obtained through the aid of the recurrence relation in (3). Proposition 1. The translated Whitney numbers of the second kind satisfy the following horizontal generating function: where Also, note that (6) can be written as where ( / ) is the falling factorial of / of order . By replacing with , we have Finally, applying the binomial inversion formula (see [4]) gives us the following explicit formula.

Proposition 2.
The translated Whitney numbers of the second kind can be expressed as Note that when = 1 in (10), we havẽ which is the known explicit formula of the Stirling numbers of the second kind. Furthermore, we have the following exponential generating function.
Proof. Multiplying both sides of (8) by / ! and summing over , gives us Now, note that The proof is completed by comparing the coefficients of ( ) in (13) and (14).

Translated Dowling Polynomials and Numbers
The well-known Bell polynomials ( ) is defined by the sum International Scholarly Research Notices 3 which consequently yields the Bell numbers when = 1. In line with this, we may define the translated Dowling polynomials as follows.
Definition 4. For nonnegative integers , , and , the translated Dowling polynomials are defined as When = 1,̃( and is called the translated Dowling numbers. Now, from (19) and (12), Hence, we have the following theorem.
Theorem 5. The following exponential generating functions hold: Remark 6. When = 1 in (22) and (23), we have which are the exponential generating functions of the classical Bell polynomials and numbers, respectively.
Sincẽ( ) ( , ) represents the number of partitions of a set with elements into subsets such that the elements of each subset can mutate in ways, except the dominant one, theñ( ) ( ) is the number of partitions of a set with elements such that the elements of each subset can mutate in ways, except the dominant one. The following theorem contains an explicit form for the polynomials̃( ) ( ; ) and numbers̃( ) ( ).

Theorem 7.
The following explicit formula holds: Proof. Combining the explicit formula in (10) with (19) yields Reindexing the sums and by further simplification, Equation (26) is obtained by letting = 1. (25) and (26), we havẽ which are the known Dobinski identities.

Convexity and Integral Representation
We will refer to (32) as convexity property. Convexity, among others, is an example of interesting global behaviours of combinatorial sequences of integers. The following theorem shows that the polynomials̃( ) ( , ) obey the convexity property.
The following beautiful integral representation of the Bell numbers was first obtained by Cesàro [12]: This expression was generalized by Mező [11] using a kind of generalization of the classical Bell numbers called -Bell numbers , . Equation (36) and Mező's identity appears to be special cases of the integral representation of the ( , )-Bell polynomials , , ( ) by R. B. Corcino and C. B. Corcino [10]. That is ,1, (1) = , and ,1,0 (1) = , respectively. The next theorem gives an integral representation for the translated Dowling polynomials. Proof. From [13], we have the following integral identity: Hence, combining this with the explicit formula in (10) yields Furthermore, we have ] sin ( ) which is the desired result.
Clearly, the integral representation in (37) boils down to Cesàro's in (36) when = 1 and = 1. Now, applying the explicit formula in (25) gives us the following. Corollary 11. The following identity holds:

More Theorems oñ( ) ( ; )
It is known that the th exponential moment of a Poisson random variable , denoted by [ ], is related to the Bell polynomials ( ) through the Dobinski's formula. That is, International Scholarly Research Notices 5 Also, the th factorial moment of with mean , denoted by [( ) ], is given by R. B. Corcino and C. B. Corcino [10] obtained a generalization of (42) using the ( , )-Bell polynomials as when = 1 and = 0. We note that identities (42), (43), and (44) can be shown to be particular cases of the generalized factorial moments by Mangontarum and Corcino [14] given by by suitable assignments of the parameters , , , and . The following lemma is analogous to (42).

Lemma 12.
The following identity holds: where is a Poisson random variable with mean .
If the mean of the Poisson random variable is / , then we have Now,̃( Using the explicit formula of the ( , )-Bell polynomials [10] , , ( ) = ( 1 ) Hence, we have the following.
Theorem 13. The following identities hold: where ( ) is the classical Dowling numbers.

The Hankel Transform of̃( ) ( )
The Hankel matrix is a matrix whose entries are symmetric with respect to the main diagonal of the matrix. It had been previously studied by some mathematicians as well as its connections in some areas of mathematics, physics, and computer science. Among these mathematicians were de Sainte-Catherine and Viennot [15], Garcia-Armas and Sethuraman [16], Tamm [17], and Vein and Dale [18]. Further theories and applications of this matrix had been established including the Hankel determinant and Hankel transform. The determinant of the Hankel matrix is called Hankel determinant, while the sequence of Hankel determinants is called Hankel transform as defined by Aigner [19]. The Hankel determinants had been previously studied by some mathematicians, for instance, Radoux [20] and Ehrenborg [21]. On the other hand, the Hankel transform was first introduced in Sloane's sequence 055878 [22] and was first studied by Layman [23]. Aigner [19] established the Hankel transform of the classical Bell numbers. A similar identity was obtained by Mező [11] for the Hankel transform of the -Bell 6 International Scholarly Research Notices numbers. In a recent paper, Corcino et al. [24] established the Hankel transform of the noncentral Bell numbers which is identical to that of the Bell and -Bell case. A more general case of Hankel transform can also be seen in [24], namely, the Hankel transform of the ( , )-Bell numbers. In this section, we are going to establish the Hankel transform of the Translated Dowling Numbers by using Aigner's method.
Using the reccurence relation in (55), we obtain This implies that Ω ( ) = Ω −1 ( ) + ( + 1) Ω ( ) With the right-hand side of (57) yields While the left hand side of (57) yields This implies that the function where ≥ 0, is a unique solution to the differential equation in (57). Hence, the exponential generating function of the th column of Λ is given by Hence, we have the following.
The next lemma is useful in establishing an identity for some matrices whose entries arẽ( ) ( ). Then for all nonnegative integers and .
Proof. By induction of , if = 0 we have which is pricisely (67).
We are now ready to state the following Hankel transform of the translated Dowling numbers.