The Noncentral Version of the Whitney Numbers : A Comprehensive Study

This paper is a comprehensive study of a certain generalization of Whitney-type and Stirling-type numbers which unifies the classical Whitney numbers, the translated Whitney numbers, the classical Stirling numbers, and the noncentral Stirling (or -Stirling) numbers. Several identities, applications, and occurrences are also presented.


Introduction
For a finite group  of order  > 0, the Dowling lattice of rank , denoted by   (), associated with  is defined by Dowling [1] as a class of geometric lattices and is known to generalize the partition lattice.Following this, Benoumhani [2] defined the Whitney numbers of the first and second kind of   (), denoted by   (, ) and   (, ), respectively, as coefficients in the expansions of the relations to Broder's [13] -Stirling numbers.However, the methods by which the former were defined appear to be of distinct motivation (cf.[13, equations (3) and (4)]).Keeping this in mind, we propose "noncentral" versions for the classical Whitney numbers parallel to the work of Koutras as seen in (11) and (12).These will serve as unified generalizations of all the abovementioned sequences of special numbers.In this comprehensive study, we present fundamental combinatorial properties such as recurrence relations, generating functions and explicit formulas, and derive more results such as the orthogonality and the inverse relations, matrix decompositions, Hankel transform, and other notable identities.Several conjectures and questions are also mentioned for further research.
Comparing coefficients of ( − )  yields the next identity useful in finding the values of w, (, ).
The next corollary can be obtained by successive application of (20).
Note that (15) can be written as w, (, )   (24) when  is replaced with  + .We are now ready to state the following proposition.
Proposition 4. The exponential generating function of the sequence {w , (, )} is given by Proof.Multiplying both sides of ( 24) by   /! and summing over  gives us Since the left-hand side is just then we have Comparing the coefficients of   completes the proof.
Theorem 5.The noncentral Whitney numbers w, (, ) satisfy the following relations: Proof.Notice that ( 30) is an obvious consequence of (29).Hence, we only choose to prove (29).Note that multiplying by   the defining relation for the Stirling numbers [   ] given by yields Comparing the coefficients of ( − )  with (16) gives the desired result.
It is known that there is no simple method in expressing first kind Stirling-type numbers explicitly.In the next theorem, we express the numbers w, (, ) in elementary symmetric polynomial form by induction.Theorem 6.The numbers w, (, ) satisfy the explicit formula International Journal of Mathematics and Mathematical Sciences Proof.Note that the theorem yields w, (0, 0) = 1, which is in line with the initial value of w, (, ) stated earlier in this section.Now, suppose the theorem holds up to  for  = 0, 1, 2, . . ., .Then by (20), Finally (33) yields w, ( + 1, ) = 1 when  =  + 1.This is in accordance with (22).

Noncentral Whitney
for any real numbers  and , nonnegative integer , and positive integer .Now, let () = ( − )  .The known difference operator yields the explicit formula Hence we propose the following combinatorial properties of the numbers W, (, ).

Proposition 7.
An explicit formula for W, (, ) is given by Moreover, the exponential generating function of the sequence { W, (, )} is given by while the horizontal generating function is Replacing  with  in (40) yields ( The next identity is also immediately obtained. It is also possible to express W, (, ) in terms of the classical Stirling numbers of the second kind.To do so, note that Using the defining relation of the Stirling numbers of the second kind, we obtain Comparing the coefficients of ()  with (40) yields Let us formally state this in the next theorem.
Theorem 10.The numbers W, (, ) satisfy the following identities: Proof.The other equality follows directly.
The next theorem can be proved by similar method used to prove (33).
where   (0) =   are the Bernoulli numbers.In relation to this, Mező [9] obtained some identities showing interesting relationships between the -Whitney numbers and the Bernoulli polynomials.The said identities are as follows: Note that when  = 1 and  = 0 in both (54) and (55), we obtain the classical identity [14] Following the same method used by Mező [9] and through the aid of the exponential generating function in (25) and the identity we propose an analogous relationship between the Bernoulli polynomials and the noncentral Whitney numbers of both kinds as follows.

Orthogonality and Inverse Relations
Proposition 13.The noncentral Whitney numbers of the first and second kind satisfy the following orthogonality relations: where is the Kronecker delta.
This proposition can be easily proved by combining ( 16) and (40).
(73) By using the "signed" translated Whitney numbers  * () (, ) [7], the matrix M ,0 can also be rewritten as (79) We are now ready to state the next theorem.
Theorem 16.The decomposition formulas of the matrices M , and N , are Proof.Since then Combining this with (71) gives us where 0 denotes an infinite-dimensional zero matrix.Consequently, because  is an arbitrary real or complex number and V 0 ( − ) is a nonzero vector, then Equation ( 81) can be shown similarly.

Noncentral Dowling and Noncentral Tanny-Dowling Polynomials
Benoumhani [2,3] was the first to introduce the following familiar polynomials: (; ) and F  (; ) are known as the Dowling and the Tanny-Dowling polynomials.Moreover, when  = 1 in (87), the resulting polynomial is called geometric polynomials [17] and was earlier studied by Tanny [18].Denoted by D, (; ) and F, (; ), the noncentral Dowling and the noncentral Tanny-Dowling polynomials can be defined as For brevity, we also call D, (; 1) ≡ D, () and F, (; 1) ≡ F, () as the noncentral Dowling and the noncentral Tanny-Dowling numbers, respectively.Notice that through the use of the exponential generating function in (39) and the explicit formula (38), the noncentral Dowling polynomials can be defined alternatively by (91) or explicitly by the Dobinski-type identity (92) in the following proposition.
Proposition 17.The following identities hold: These identities are actually equivalent to those of the -Whitney polynomials when  = − and are generalizations of the translated Dowling polynomials (cf.[6, equations (22) and (25)]).As for the noncentral Tanny-Dowling polynomials, since (39) can be rewritten as then we get This is equivalent to (95) in the next theorem.
Theorem 18.The polynomials F, (; ) satisfy the exponential generating function The case in (95), where  = , immediately yields Proof.To prove this theorem, we first show that Note that by algebraic manipulation, one readily gets Reindexing the third sum and using (96), we get Comparing the coefficients of   yields (98).The proof is then completed when  is replaced with  in (98).
Identity (98) used in the proof of the previous theorem is a generalization of Benoumhani's [3,Theorem 4].On the other hand, when  = 1, Moreover, we get the familiar representation of F  (; 1) due to Rota [19] given by when  = −1 in (101).
It is well-known that the binomial coefficients (   ) satisfy the binomial inversion formula The rest of this section contains corollaries which are obtained through the use of this.
Theorem 20.The noncentral Whitney numbers of the second kind satisfy the following recursion formula: Proof.Using the explicit formula (38) gives us Reindexing the summation yields (104).
The next corollary is easily obtained by applying binomial inversion formula to (113).(120) The binomial inversion formula readily yields the following.
The next theorem and corollary can be obtained by similar method as the previous ones.The proof is left as exercise.( The th Bell polynomial is known to satisfy the explicit formula Corollary 32.The Bell polynomials satisfy the following identity: The case where  = −1 in ( 127) and ( 128) is due to Rahmani (cf.(130)

The Hankel Transform of Noncentral Dowling Numbers
Hankel matrices had been studied by several mathematicians because of their connections in some areas of mathematics, physics, and computer science.Further theories and applications of these matrices have been established including the Hankel determinant and Hankel transform.The Hankel transform was first introduced in Sloane's sequence 055878 [21] and was later on studied by Layman [22].Layman [22] first defined the Hankel transform of an integer sequence as the sequence of Hankel determinants of order  of a given sequence.Among the remarkable properties established by Layman [22] is the property that any integer sequence has the same Hankel transform as its binomial transform, as well as its invert transform.In this section, we thoroughly investigate the Hankel transform of the noncentral Dowling numbers using this property.Let Γ = ( , ) be the infinite lower triangular matrix defined recursively by where  ≥ 1,  0,0 = 1,  0, = 0 if  > 0 and  , = 0 if  < .
The next proposition shows that (131) is a recurrence relation of the infinite lower triangular matrix Γ = ( , ), where the entries in the 0-column are the numbers D,0 ().Proposition 34.Let Φ  () be the exponential generating function of the th column of matrix Γ.Then where  ≥ 0 and the 0-column entries of Γ are the numbers D,0 ().
To obtain the Hankel transform of the noncentral Dowling numbers, the next lemma which may be proved by induction is essential.
and {D , } be the Hankel transform of the numbers D, ().
Proof.Suppose  −  ≥ 0. Then This is precisely the desired result.
Cesàro [27] obtained an integral representation of the Bell numbers   fl   (1), namely, Several generalizations of this remarkable representation were presented by Mező [24], Mangontarum et al. [6], and R. B. Corcino and C. B. Corcino [25].To establish an analogous representation for D, (; ), we take the explicit formula in (38) and substitute it to the right-hand side of Callan's [28] integral identity given by That is, we obtain Multiplying both sides by   and summing over  give where   (, ) is an extension of Bell polynomials satisfying It is also known that the th factorial moment of  is Now, if we take the expectation of (40), we get through the aid of (38).Reindexing the sum yields Clearly, when  is replaced with /, On the other hand, using the binomial theorem, The above results are compiled in the next theorem.
Theorem 45.The following identities hold:

Some Questions and Conjectures
There are a number of further applications and possible extensions of the numbers introduced in this paper.The authors would like to direct the attention of the readers to some questions and conjectures.Several studies regarding the identification of the index for which certain Stirling-type numbers attain their maximum value were conducted earlier by some mathematicians, for instance, Mező [24,31] for the -Stirling and -Bell numbers, R. B. Corcino and C. B. Corcino [32] for the generalized Stirling numbers, and recently, Corcino et al. [33] for the noncentral Stirling numbers of the first kind.Question 1.Is it possible to identify the maximizing index of the noncentral Whitney numbers of both kinds?Will these be different from the said earlier results?Perhaps this question may be answered by the so-called "Erdős and Stone Theorem" mentioned in [33].
The study of asymptotic estimates/approximations and asymptotic formulas for Stirling-type numbers (such as the (, )-Stirling numbers and the -Whitney numbers of the second kind) has been the interest of several mathematicians, especially C. B. Corcino and R. B. Corcino [34,35], Corcino et al. [36], and Corcino et al. [37].The next question is an interesting motivation for further study.Question 2. It is compelling to study asymptotic approximations and obtain formulas for the noncentral Whitney numbers.However, will these formulas be distinct from those results found in [34][35][36][37] or will they be equivalent?Corcino et al. [38] defined distinct "-analogues" for the noncentral Stirling numbers of the second kind.Using Definition 47, the said -noncentral Stirling numbers of the second kind were given a combinatorial interpretation in the context of -tableaux.A more general study can actually be seen in the work of de Médicis and Leroux [39].
Definition 47 (see [39]).An -tableau is a list Φ of columns  of Ferrer's diagram of a partition Λ (by decreasing order of length) such that the lengths || are part of the sequence  = (  ) ≥0 , a strictly increasing sequence of nonnegative integers.
The key to proving this remarkable observation might be achieved using the explicit formula in (52).
A "multiparameter version" of the noncentral Stirling numbers of Koutras [12] was introduced by El-Desouky [40].This extended the number of parameters from the usual one that is  to a sequence   ,  = 0, 1, . . .,  −1 .analogues of these numbers were then defined by Corcino and Mangontarum [41].

Conjecture 49. It is possible to establish a multiparameter version of the noncentral Whitney numbers of both kinds (may be called multiparameter noncentral Whitney numbers) either by means of a triangular recurrence relation or by a certain generating function.
Before ending this section, let us first note that Mangontarum and Katriel [42] investigated a connection of the defining relations of the -Whitney numbers of both kinds in (9) and (10)  (174) Using their observations in this matter, they were able to define a remarkable -deformation of the -Whitney numbers using the -Boson operators of Arik and Coon [43] which satisfies

Corollary 28 .
The noncentral Dowling and Tanny-Dowling polynomials satisfy the relations given by