# The Translated Dowling Polynomials and Numbers

Authors: Mahid M. Mangontarum; Amila P. Macodi-Ringia; Normalah S. Abdulcarim
Journal: International Scholarly Research Notices
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

In 1996, the classicalWhitney numbers of the second kindWm(n,k)of Dowling Lattices was introduced by Benoumhani [1]. Wm(n,k) satisfy the recurrence relation:
(1)Wm(n,k)=Wm(n-1,k-1)+(1+mk)Wm(n-1,k).
Other fundamental properties of these numbers were already established by Benoumhani in [1, 2]. The numbers Wm(n,k) can be shown to be a kind of generalization of the famousStirling numbers of the Second kindS(n,k) when the parameter m equals to 1. That is,
(2)W1(n,k)=S(n,k).
Recently, a translated version of the Whitney numbers of the second kind was introduced by Belbachir and Bousbaa [3] which they namedtranslated Whitney numbers of the second kind, denoted by W~(α)(n,k). W~(α)(n,k) actually counts the number of partitions of a set with n elements into k subsets such that the elements of each subset can mutate in α ways, except the dominant one. To compute the first few values of these numbers, the following recurrence relation was obtained in [3]:
(3)W~(α)(n,k)=W~(α)(n-1,k-1)+kαW~(α)(n-1,k).
The classical Stirling numbers of the second kind can also be obtained from these numbers when α=1. On the otherhand, the classicalDowling numbersDm(n) are defined to be the sum of  Wm(n,k). That is,
(4)Dm(n)=∑k=0n‍Wm(n,k)
and can be computed using the explicit formula:
(5)Dm(n)=1e1/m∑k≥0‍mk+1nmkk!.Dm(n) is known to be a generalization of the classicalBell numbers which is the sum of the Stirling numbers of the second kind S(n,k). In this paper, we will define thetranslated Dowling numbers as the sum of W~(α)(n,k). The content of this paper is summarized as follows. In Section 2, we will introduce some basic properties for the numbers W~(α)(n,k). 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.


## 2. Some Properties ofW~(α)(n,k)

Interesting properties ofW~(α)(n,k) can also be obtained parallel to those done in [1]. For instance, by induction on n, 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:(6)tn=∑k=0n‍W~α(n,k)t∣αk,
where t∣αk=∏i=0k-1‍t-iα is the generalized factorial of t of increment α.Also, note that (6) can be written as
(7)tn=∑k=0n‍W~(α)(n,k)αktαk,
where t/αk is the falling factorial of t/α of order k. By replacing t with kα, we have
(8)αkn=∑j=0n‍W~(α)(n,j)αjkj,=∑j=0n‍kjW~(α)(n,j)αjkjkj.
Finally, applying the binomial inversion formula (see [4])
(9)fk=∑j=0k‍kjgj⟺gk=∑j=0k‍-1k-jkjfj
gives us the following explicit formula.Proposition 2.
The translated Whitney numbers of the second kind can be expressed as(10)W~(α)(n,k)=1αkk!∑j=0k‍-1k-jkjαjn.Note that whenα=1 in (10), we have
(11)W~1n,k=1k!∑j=0k‍-1k-jkjjn=S(n,k),
which is the known explicit formula of the Stirling numbers of the second kind. Furthermore, we have the following exponential generating function.Proposition 3.
The numbersW~(α)(n,k) satisfy
(12)∑n≥k‍W~(α)(n,k)znn!=1k!eαk-1αk.Proof.
Multiplying both sides of (8) by zn/n! and summing over n, gives us
(13)∑n≥0‍αtnznn!=∑n≥0‍∑k=0n‍W~αn,kαktkznn!=∑k≥0‍αk∑n≥k‍W~(α)(n,k)znn!tk.
Now, note that
(14)∑n≥0‍αtnznn!=1+eαz-1t=∑k=0t‍tkeαz-1k=∑k=0t‍eαz-1kk!tk.
The proof is completed by comparing the coefficients of (t)k in (13) and (14).In 2010, Mező [5], introduced the r-Whitney numbers of the second kind Wm,r(n,k) as coefficients in the expansion of
(15)mx+rn=∑k=0n‍mkWm,r(n,k)xk.
These numbers actually are equivalent to the (r,β)-Stirling numbers nkr,β defined by Corcino et al. [6]. That is,
(16)Wβ,r(n,k)=nkr,β.
Moreover, we have
(17)W1,0(n,k)=S(n,k),W1,r(n,k)=n+rk+rr,Wm,0(n,k)=Wm(n,k),Wα,0(n,k)=W~(α)(n,k),
where n+rk+rr is the r-Stirling numbers of the second kind by Broder [7]. This means that the identities (6), (10), and (12) for the numbers W~(α)(n,k) appear to be special cases of Wm,r(n,k) (see [5, 8]).


## 3. Translated Dowling Polynomials and Numbers

The well-knownBell polynomialsBn(x) is defined by the sum
(18)Bn(x)=∑k=0n‍S(n,k)xk
which consequently yields the Bell numbers Bn when x=1. In line with this, we may define the translated Dowling polynomials as follows.Definition 4.
For nonnegative integersn, k, and α, the translated Dowling polynomials are defined as
(19)D~(α)(n;x)=∑k=0n‍W~(α)(n,k)xk.
When x=1,
(20)D~(α)(n;1)=D~(α)(n)=∑k=0n‍W~(α)(n,k)
and is called the translated Dowling numbers.Now, from (19) and (12),
(21)∑n≥0‍D~αn;xznn!=∑n≥0‍∑k=0n‍W~(α)(n,k)znn!xk=∑k≥0‍eαz-1kk!αkxk=ex(eαz-1)/α.
Hence, we have the following theorem.Theorem 5.
The following exponential generating functions hold:(22)∑n≥0‍D~(α)(n;x)znn!=exp⁡xezα-1α;(23)∑n≥0‍D~(α)(n)znn!=exp⁡ezα-1α.Remark 6.
Whenα=1 in (22) and (23), we have
(24)∑n≥0‍D~(1)(n;x)znn!=exez-1;∑n≥0‍D~(1)(n)znn!=eez-1,
which are the exponential generating functions of the classical Bell polynomials and numbers, respectively.SinceW~(α)(n,k) represents the number of partitions of a set with n elements into k subsets such that the elements of each subset can mutate in α ways, except the dominant one, then D~(α)(n) is the number of partitions of a set with n elements such that the elements of each subset can mutate in α ways, except the dominant one. The following theorem contains an explicit form for the polynomials D~(α)(n;x) and numbers D~(α)(n).Theorem 7.
The following explicit formula holds:(25)D~(α)(n;x)=1ex/α∑i≥0‍iαni!xαi;(26)D~(α)(n)=1e1/α∑i≥0‍iαni!αi.Proof.
Combining the explicit formula in (10) with (19) yields
(27)D~(α)(n;x)=∑j≥0‍∑k≥j‍-1jkjk-jnk!αn-kxk.
Reindexing the sums and by further simplification,
(28)D~αn;x=∑j≥0‍-xjαjj!∑i≥0‍inxiαii!αn=e-x/α∑i≥0‍iαni!xαi.
Equation (26) is obtained by letting x=1.Remark 8.
Whenα=1 in (25) and (26), we have
(29)D~(1)(n;x)=1ex∑i≥0‍ini!xi=Bn(x);D~(1)(n)=1ex∑i≥0‍ini!=Bn,
which are the known Dobinski identities.To close this section, we will cite ther-Dowling polynomials Dm,r(n,x) of Cheon and Jung [8] defined by
(30)Dm,r(n,x)=∑k=0n‍Wm,r(n,k)xk.
Properties of Dm,r(n,x) were already established in [8] and were further studied by Rahmani [9]. We note that the polynomials Dm,r(n,x) coincide with the (r,β)-Bell polynomials Gn,β,r(x) of R. B. Corcino and C. B. Corcino [10]. That is, Dβ,r(n,x)=Gn,β,r(x). Moreover,
(31)D1,0(n,x)=Bn(x),D1,r(n,x)=Bn,r(x),Dα,0(x)=D~(α)(n;x),
where Bn,r(x) is the r-Bell polynomials in [11].


## 4. Convexity and Integral Representation

A real sequencevk, k=0,1,2,… is calledconvex [4] on an interval [a,b], where [a,b] contains at least 3 consecutive integers, if
(32)vk≤12vk-1+vk+1,k∈[a+1,b-1].
We will refer to (32) asconvexity property. Convexity, among others, is an example of interesting global behaviours of combinatorial sequences of integers. The following theorem shows that the polynomials D~(α)(n,x) obey the convexity property.Theorem 9.
Letx>0 and α≥0. Then
(33)D~(α)(n+1;x)≤12D~(α)(n;x)+D~(α)(n+2;x)
for n=1,2,3,….Proof.
Sinceαk≥0, then
(34)0≤1-αk20≤1-2(αk)+αk22αk≤1+αk22αkn+1≤αkn+αkn+2αkn+1≤12αkn+αkn+2.
Multiplying both sides by x/αk(1/k!) and summing over i yields
(35)∑i≥0‍αkn+1i!xαi≤12∑i≥0‍αkni!xαi+∑i≥0‍αkn+2i!xαi.
Finally, multiplying both sides by e-x/α and using (25) completes the proof.The following beautiful integral representation of the Bell numbersBn was first obtained by Cesàro [12]:
(36)Bn=2n!πeIm⁡∫0π‍eeeiθsin(nθ)dθ.
This expression was generalized by Mező [11] using a kind of generalization of the classical Bell numbers called r-Bell numbers Bn,r. Equation (36) and Mező’s identity appears to be special cases of the integral representation of the (r,β)-Bell polynomials Gn,β,r(x) by R. B. Corcino and C. B. Corcino [10]. That is Gn,1,r(1)=Bn,r and Gn,1,0(1)=Bn, respectively. The next theorem gives an integral representation for the translated Dowling polynomials.Theorem 10.
The translated Dowling polynomials have the following integral representation:(37)D~(α)(n;x)=2n!πex/αIm⁡∫0π‍exp⁡xαeαeiθsin(nθ)dθ,
where i=-1.Proof.
From [13], we have the following integral identity:
(38)Im⁡∫0π‍ejeiθsin(nθ)dθ=π2jnn!.
Hence, combining this with the explicit formula in (10) yields
(39)π21n!W~αn,k=1αkk!∑j=0k‍-1k-jkjπ2αjnn!=1αkk!∑j=0k‍-1k-jkjiiii×Im⁡∫0π‍eαjeiθsin⁡(nθ)dθ=1αkk!×Im⁡∫0π‍∑j=0k‍-1k-jkjeαeiθjsin⁡nθdθ=Im⁡∫0π‍eαeiθ-1kαkk!sin(nθ)dθ.
Furthermore, we have
(40)∑k≥0‍W~αn,kxk=2n!πIm⁡∫0π‍∑k≥0‍eαeiθ-1kk!xαksin⁡nθdθ=2n!πex/αIm⁡∫0π‍exp⁡xαeαeiθsin(nθ)dθ,
which is the desired result.Clearly, the integral representation in (37) boils down to Cesàro’s in (36) when α=1 and x=1. Now, applying the explicit formula in (25) gives us the following.Corollary 11.
The following identity holds:(41)∑j≥0‍jαnj!xαj=2n!πIm⁡∫0π‍exp⁡xαeαeiθsin(nθ)dθ.


## 5. More Theorems onD~(α)(n;x)

It is known that thenth exponential moment of a Poisson random variable X, denoted by Eλ[Xn], is related to the Bell polynomials Bn(λ) through the Dobinski’s formula. That is,
(42)Eλ[Xn]=Bn(λ).
Also, the nth factorial moment of X with mean λ, denoted by Eλ[(X)n], is given by
(43)Eλ[Xn]=λn.
R. B. Corcino and C. B. Corcino [10] obtained a generalization of (42) using the (r,β)-Bell polynomials as
(44)Eλ/ββX+rn=Gn,β,r(X)
when β=1 and r=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
(45)EλβX+γ∣αn=e-λ∑i≥0‍iβ+γ∣αni!λi,Eλ(αX-γ∣β)n=e-λ∑i≥0‍iα-γ∣βni!λi,
by suitable assignments of the parameters α, β, γ, and λ. The following lemma is analogous to (42).Lemma 12.
The following identity holds:(46)EλαXn=1eλ∑j≥0‍αjnj!λj,
where X is a Poisson random variable with mean λ.Proof.
From (8),
(47)αXn=∑k=0n‍W~(α)(n,k)αkXk.
Hence by (43),
(48)EλαXn=Eλ∑k=0n‍W~(α)(n,k)αkXk=∑k=0n‍W~(α)(n,k)αkEλXk=∑k=0n‍W~(α)(n,k)αkλk.
Using the explicit formula in (10) and simplifiying further completes the proof.If the mean of the Poisson random variableX is λ/α, then we have
(49)Eλ/ααXn=D~(α)(n;λ).
Now,
(50)D~αn;λ=Eλ/α-1+αX+1n=∑k=0n‍nk-1n-kEλ/ααX+1k=∑k=0n‍nk-1n-kGk,α,1(λ).
Using the explicit formula of the (r,β)-Bell polynomials [10]
(51)Gn,β,r(x)=1ex/β∑k≥0‍x/βkk!βk+rn
yields
(52)D~(α)(n;λ)=∑k=0n‍nk-1n-k1eλ/α∑j≥0‍λ/αjj!αj+1k.
Hence, we have the following.Theorem 13.
The following identities hold:(53)D~(α)(n;λ)=∑k=0n‍nk-1n-k1eλ/α∑j≥0‍αj+1kαjj!λj;D~(α)(n;1)=∑k=0n‍nk-1n-kDα(k),
where Dα(k) is the classical Dowling numbers.The next theorem is easily deduced from (22) and (46).Theorem 14.
The following exponential generating functions hold:(54)∑n≥0‍EλαXnznn!=e(eαz-1)λ;∑n≥0‍Eλ/ααXnznn!=∑n≥0‍D~(α)(n;x)znn!;∑n≥0‍E1/ααxnznn!=∑n≥0‍D~(α)znn!.


## 6. The Hankel Transform ofD~(α)(n)

TheHankel 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 theHankel determinant andHankel 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 A055878 [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 r-Bell 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 r-Bell case. A more general case of Hankel transform can also be seen in [24], namely, the Hankel transform of the (r,β)-Bell numbers. In this section, we are going to establish the Hankel transform of the Translated Dowling Numbers by using Aigner’s method.LetΛ=(am,k) be the infinite lower triangular matrix defined recursively by
(55)am,k=am-1,k-1+αk+1am-1,k+αk+1am-1,k+1,
where m≥1, a0,0=1, a0,k=0 if k>0, and am,k=0 if m<k.Using the reccurence relation in (55), we obtain
(56)∑m=0∞‍am,kzm-1(m-1)!=∑m=0∞‍am-1,k-1zm-1m-1!+αk+1×∑m=0∞‍am-1,kzm-1(m-1)!+α(k+1)∑m=0∞‍am-1,k+1zm-1(m-1)!.This implies that(57)Ωk′z=Ωk-1(z)+(αk+1)Ωk(z)+α(k+1)Ωk+1(z).With(58)Ωk(z)=e(eαz-1)/αeαz-1kαkk!,
the right-hand side of (57) yields
(59)Ωk-1(z)+(αk+1)Ωk(z)+α(k+1)Ωk+1(z)=e(eαz-1)/αeαz-1k-1αk-1k-1!+(αk+1)e(eαz-1)/αeαz-1kαkk!+α(k+1)e(eαz-1)/αeαz-1k+1αk+1(k+1)!=e(eαz-1)/αeαz-1k-1αk-1k-1!+e(eαz-1)/αeαz-1k-1(eαz-1)αk-1k-1!+e(eαz-1)/αeαz-1kαkk!+e(eαz-1)/αeαz-1k(eαz-1)αkk!=e(eαz-1)/αeαz-1k-1αk-1k-1!+e(eαz-1)/αeαz-1k-1eαzαk-1(k-1)!-e(eαz-1)/αeαz-1k-1αk-1k-1!+e(eαz-1)/αeαz-1kαkk!+e(eαz-1)/αeαz-1keαzαkk!-e(eαz-1)/αeαz-1kαkk!=e(eαz-1)/α+αzeαz-1k-1αk-1k-1!+e(eαz-1)/α+αzeαz-1kαkk!.While the left hand side of (57) yields
(60)Ωk′z=e(eαz-1)/αeαz-1k-1αkk!kαeαz+e(eαz-1)/αeαz-1kαkk!1ααeαz=e(eαz-1)/α+αzeαz-1k-1αk-1k-1!+e(eαz-1)/α+αzeαz-1kαkk!.This implies that the function(61)e(eαz-1)/αeαz-1kαkk!,
where k≥0, is a unique solution to the differential equation in (57). Hence, the exponential generating function of the kth column of Λ is given by
(62)Ωk(z)=e(eαz-1)/αeαz-1kαkk!.
Hence, we have the following.Lemma 15.
LetΩk(z) be the exponential generating function of the kth column of matrix Λ. That is,
(63)Ωk(z)=∑m=0∞‍am,kzmm!.
Then
(64)Ωk(z)=e(eαz-1)/α·eαz-1kαkk!,
where k≥0 and Ω0(z)=∑m=0∞‍D~αmzm/m!. That is, the 0-column entries of Λ are the numbers D~(α)(m), m=0,1,2,3….Remark 16.
Whenk=0 in (64), we have
(65)Ω0(z)=exp⁡eαz-1α
which is the exponential generating function in (23).The next lemma is useful in establishing an identity for some matrices whose entries areD~(α)(m).Lemma 17.
Letsm be the mth row of Λ=(am,k). Define
(66)sm∘sn=∑k≥0‍am,kan,kαkk!.
Then
(67)sm∘sn=am+n,0=D~(α)(m+n),
for all nonnegative integers m and n.Proof.
By induction ofm, if m=0 we have
(68)s0∘sn=∑k≥0‍a0,kan,kαkk!.
Since a0,k=0∀k>0,
(69)s0∘sn=a0,0an,0α00!=a0+n,0∀n≥0.
Suppose that sl∘sn=al+n,0 holds for l≤m-1 and all n. Then by (55)(70)sm∘sn=∑k≥0‍am,kan,kαkk!=∑k≥0‍am-1,k-1+αk+1am-1,kiiiiiiiii+α(k+1)am-1,k+1an,kαkk!=∑k≥0‍am-1,k-1an,kαkk!+∑k≥0‍αk+1am-1,kan,kαkk!+∑k≥0‍α(k+1)am-1,k+1an,kαkk!.
Reindexing the summation yields
(71)sm∘sn=∑k≥-1‍am-1,kan,k+1αk+1(k+1)!+∑k≥0‍(αk+1)am-1,kan,kαkk!+∑k≥1‍αkam-1,kan,k-1αk-1k-1!=∑k≥0‍am-1,kan,k+1αk+1k+1!+∑k≥0‍αk+1am-1,kan,kαkk!+∑k≥0‍am-1,kan,k-1αkk!=∑k≥0‍an,k-1+αk+1an,kiiiiiiiii+αk+1an,k+1am-1,kαkk!.
By (55),
(72)sm∘sn=∑k≥0‍an+1,kam-1,kαkk!.
From the inductive hypothesis,
(73)sm∘sn=a(n+1)+(m-1),0=an+m,0=D~(α)(m+n),
which is pricisely (67).We are now ready to state the following Hankel transform of the translated Dowling numbers.Theorem 18.
The numbersD~(α)(m) have the Hankel Transform
(74)D~(α)(0)D~(α)(1)D~(α)(2)⋯D~(α)(m)D~(α)(1)D~(α)(2)D~(α)(3)⋯D~(α)(m+1)⋮⋮⋮⋯⋮D~(α)(m)D~(α)(m+1)D~(α)(m+2)⋯D~(α)(2m)=∏r=0m‍αrr!=αm+12r!!.Proof.
LetΛm be the lower triangular submatrix of Λ consisting of the rows and columns numbered 0 to m. Then Λm is a matrix with diagonal 1. It follows that det⁡Λm=1. This implies the determinant of the transpose of Λm is one; that is, det⁡ΛmT=1. Let Λ^m=αrr!ai,r0≤i,r≤m. Then
(75)det⁡Λ^m=∏r=0m‍αrr!.
By (67),
(76)Λ^m·ΛmT=bi,r0≤i,r≤m,
where bi,r=∑k=0m‍ai,kar,kαrr!=ai+r,0=D~(α)(i+r). That is,
(77)Λ^m·ΛmT=D~αi+r0≤i,r≤m.
Thus,
(78)det⁡Λ^m·ΛmT=(det⁡Λ^m)(det⁡ΛmT)=∏r=0m‍αrr!.
This is the desired result.Remark 19.
Note that whenα=1, we recover from (74) the Hankel transform of the classical Bell numbers of Aigner [19], the Hankel transform of the r-Bell numbers of Mező [11], and the Hankel transform of the noncentral Bell numbers in [24]. This makes (74) a generalization of the previously mentioned Hankel transforms. Also, the Hankel transform of the (r,β)-Bell numbers [24] appears to be analogous to the Hankel transform of the translated Dowling numbers in (74).Much is yet to be learnt regarding the translated Dowling polynomials and numbers. It is interesting to establish more properties for these polynomials and numbers parallel to the properties of the Bell polynomials and numbers, and their generalizations. The authors also recommend further study regarding the translatedr-Whitney numbers [3]. The results in this paper might be extended to translate r-Dowling polynomials and numbers using the translated r-Whitney numbers of the second kind. Another interesting topic can be found in [25] where Corcino et al. obtained the asymptotic formulas for the r-Whitney numbers of the second kind as well as the range of validity of each formula. It would be compelling to do the same to the translated Whitney numbers of the second kind.



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